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omar furTuxia,

zurab cigroSvili,

qeTevan manjgalaZe

albaToba da maTematikuri

statistika

(leqciebis kursi qimiis, biologiis da sicocxlis

Semswavleli mecnierebebis mimarTulebebisstudentebisaTvis)

T s u

2

leqciebis kursSi gadmocemulia Tanamedrove albaTobis Teoriisa

da maTematikuri statistikis ZiriTadi cnebebi da faqtebi, romlebic

aucilebelia rTuli qimiuri da biologiuri procesebis Sesaswavlad

da maTTvis met-naklebad adekvaturi modelebis asagebad. masSi warmod-genilia maTematikuri statistikis iseTi nawilebi, rogoricaa Sefaseb-

isa da hipoTezaTa Semowmebis Teoria erTi da or amokrefiani amocane-

bisaTvis, dispersiuli da regresiul-korelaciuri analizi, kategori-

uli tipis monacemebis analizi da statistikis araparametruli meTo-

debi.

leqciebis kursi gankuTvnilia iv. javaxiSvilis saxelobis Tbilis-

is saxelmwifo universitetis, qimiis, biologiisa da sicocxlis Sems-

wavlel mecnierebaTa mimarTulebebis studentebisaTvis; igi sasargeb-

lo iqneba sxva umaRlesi saswavleblebis (magaliTad, samedicino da

teqnikuri universitetis Sesabamisi mimarTulebebis) studentebisa dastatistikis gamoyenebiTi aspeqtebiT dainteresebuli sxvadasxva pro-

filis specialistTa farTo wrisaTvis.

3

winasityvaoba

winamdebare leqciebis kursi albaTobasa da maTematikur stati-

stikaSi (qimiis, biologiisa da sicocxlis Semswavlel mecnierebaTa

mimarTulebebis meoTxe kursis studentebisaTvis) warmoadgens am mima-

rTulebaTa studentebisaTvis gankuTvnili umaRlesi maTematikis sagnis

Semadgeneli sami kursidan (kalkulusi, diferencialuri modelebi,

albaToba da maTematikuri statistika) mesame nawils. is efuZneba

2007-2009 wlebSi aRniSnuli mimarTulebebis (da agreTve socialur

da politikur mecnierebaTa da ekonomikisa da biznesis mimarTulebeb-

is) studentebisTvis albaTobis Teoriasa da maTematikur statistika-Si wakiTxul leqciebs. misi Sedgenisas, garda iv. javaxiSvilis saxel-

obis Tbilisis saxelmwifo universitetSi am kuTxiT arsebuli gamoc-

dilebisa, gamoyenebulia Sesabamisi kursis harvardis samedicino saja-

ro skolaSi (aSS) wakiTxvis gamocdileba. leqciebis kurss Tan erTvis

qarTul da inglisur enebze silabusi saTanado ZiriTadi da damxmare

literaturiT da leqciebis mixedviT paragrafebis miTiTebiT. leqcieb-

is kursi sasargeblo iqneba rogorc iv. javaxiSvilis saxelobis Tbi-

lisis saxelmwifo universitetis, aseve qimiis, biologiis da sicocx-

lis Semswavlel mecnierebaTa mimarTulebebis sxva umaRlesi saswavle-

blebis studentebisa da magistrantebisaTvis.

o. furTuxia

Tbilisi, marti,

2009

4

Sesavali

albaTobis Teoria warmoadgens maTematikis dargs, romelic Sei-

swavlis iseTi eqsperimentebis (movlenebis) maTematikur modelebs,

romelTa Sedegebi calsaxad ar ganisazRvreba cdis pirobebiT. aseT eq-

sperimentebs ewodebaT SemTxveviTi eqsperimentebi. SemTxveviTi eqsperi-mentebia: monetis agdebis Sedegi, mizanSi srolisas miznis dazianeba an

ardazianeba, xelsawyos muSaobis xangrZlivoba, satelefono sadgurSi

gamoZaxebaTa ricxvi, avto-sagzao SemTxvevaTa raodenoba, arabejiTi

studentis mier gamocdis Cabarebis Sedegi, valutis kursi rom ar Ca-

mova garkveul niSnulamde da sxva.

saWiroa aRiniSnos, rom albaTobis Teoria ikvlevs ara nebismi-

er SemTxveviT eqsperiments, aramed mxolod iseT eqsperimentebs, rom-

lebic xasiaTdebian statistikuri mdgradobis anu sixSireTa mdgrado-

bis TvisebiT: sixSireebi in nAi

/)( (sadac in i -ur seriaSi eqsperimen-

tebis ricxvia, xolo )(Ain _ ki A xdomilebis moxdenaTa raodeno-

ba am seriaSi) axlos arian erTmaneTTan (mcired gansxvavdebian erTman-

eTisagan an raime saSualo sidididan) yovelTvis rogorc ki in ricx-

vebi iqnebian sakmaod didebi. magaliTad, v. feleris wignSi moyvanilia

monetis agdebis 10 seriaSi ( 10,...1i ), sadac TiTeul seriaSi

1000in eqsperimentia, gerbis mosvlaTa )"("gin ricxvis Semdegi mo-

nacemebi: 501, 485, 509, 536, 485, 488, 500, 497, 494, 484. cxadia,

rom aq fardobiTi sixSireebi axlosaa erTmaneTTan, da maSasadame, eqs-

periments, romelic mdgomareobs simetriuli monetis agdebaSi, gaaCnia

sixSiris mdgradobis Tviseba.g. krameris mier moyvanili monacemebi 1935 wels SveciaSi daba-

debuli axalSobilebis Sesaxeb (sadac n _ axalSobilTa raodenobaa,

xolo /m n _ vaJebis dabadebis fardobiTi sixSire) ase gamoiyureba:

Tveebi I II III IV V VIn 7280 6957 7883 7884 7892 7609

m/n 0.515 0.510 0.510 0.529 0.522 0.518Tveebi VII VIII IX X XI XII sul

n 7585 7393 7203 6903 6552 7132 88273m/n 0.523 0.514 0.515 0.509 0.518 0.527 0.517

miuxedavad imisa, rom axalSobilTa saerTo raodenoba icvleba

wlis ganmavlobaSi, vaJebis dabadebis fardobiTi sixSire sakmaod

mdgradad meryeobs 0.517 – saSualo mniSvnelobis irgvliv.

analogiuri statistikuri kanonzomierebebi aRmoCenil iqna me-18

saukuneSi demografiul monacemebSi – axalSobilTa statistikis Sesw-

5

avlisas, sikvdilianobis statistikis Seswavlisas, ubdur SemTxvevaTastatistikis Seswavlisas da a. S. (rac, Tavis mxriv, sakmaod efeqtur-

ad gamoiyebneboda sadazRvevo kompaniebis saqmianobaSi). mogvianebiT, me-

19 saukunis bolos da me-20 saukunis dasawyisSi axali statistikuri

kanonzomierebebi aRmoCenil iqna fizikaSi, qimiaSi, biologiaSi, sicoc-

xlis Semswavlel mecnierebebSi, ekonomikaSi da sxva mecnierebebSi. am

kanonzomierebebs mivyavarT albaTobis statistikuri gansazRvrisaken:

ricxvs, romlis irgvlivac irxeva A xdomilebis fardobiTi sixSire,

ewodeba A xdomilobis albaToba da aRiniSneba )(AP simboloTi. maT-

ematikuri dasabuTeba imisa, rom fardobiTi sixSire axlosaa albaTob-

asTan moyvanilia iakob bernulis cnobil TeoremaSi, romelic albaTo-

bis TeoriaSi cnobilia agreTve did ricxvTa kanonis saxelwodebiT.

aRsaniSnavia, rom yvela SesaZlo eqsperimenti SeiZleba gaiyos

sam kategoriad: I. eqsperimentebi sruli mdgradobiT, sadac saerTod

araa ganuzRvreloba; II. eqsperimentebi sadac ara gvaqvs sruli mdgra-

doba, magram aris statistikuri mdgradoba; III. eqsperimentebi sadac

statistikuri mdgradobac ki ara gvaqvs.pirvel jgufs miekuTvneba umravlesoba eqsperimentebis, romleb-

ic aRiwerebian sabunebismetyvelo mecnierebebis (fizika, qimia, biolo-

gia) klasikuri kanonebiT da maTi Seswavla xdeba albaTobis Teoriis

gamoyenebis gareSe. mesame kategoriisaTvis albaTobis Teoria gamousad-

egaria. meore jgufi warmoadgens swored albaTobis Teoriis gamoyene-

bis ares. TviTon albaTobis Teoriis mier gadasawyveti amocanebi iyo-

fa or did jgufad. pirveli jgufis amocanebs SeiZleba vuwodoT amo-

canebi rTuli xdomilebebis albaTobebis gamoTvlaze, roca cnobilia

martivi xdomilebebis albaTobebi.

amocanebis meore jgufi garkveuli azriT pirveli jgufis amo-canebis Sebrunebulia. aq eqsperimentebis seriis Sedegebis safuZvelze

raime gziT unda SevafasoT martivi xdomilebebis albaTobebi. vinaidan,

martivi xdomilebebis albaTobebi ucnobia im amocanebSi, romlebic

praqtikaSi warmoiSoba, amitom amocanebis es jgufi gansakuTrebiT sai-

nteresoa gamoyenebebis TvalsazrisiT. albaTobis Teoriis im nawils,

romelic axdens meore gjufis amocanebis zust dasmas da ikvlevs

maTi amoxsnis meTodebs, maTematikuri statistika ewodeba.

maTematikuri statistika – es aris maTematikis dargi, romelic

Seiswavlis statistikuri monacemebis sistematizaciis, analizisa da

gamoyenebis meTodebs Teoriuli da praqtikuli daskvnebis miRebis miz-niT. igi Sedgeba ori nawilisagan: aRweriTi (deskrifciuli) statist-

ika da statistikuri daskvnebis Teoria. misi meTodebi albaTobis Teo-

riazea dafuZnebuli. albaTobis Teoria warmoadgens xids aRweriT

statistikasa da statistikuri daskvnebis Teorias Soris, igi saSua-

lebas iZleva gaviazroT, Tu rogor iqmneba da gamoiyeneba statistiku-

6

ri procedurebi, rogor SeiZleba aviciloT Tavidan am meTodebis ara-swori gamoyeneba.

albaTobis Teoriis rogorc mecnierebis ganviTareba miekuTvneba

me-17 saukunis meore naxevars da dakavSirebulia paskalis (1623-1662),

fermas (1601-1665) da hiugensis (1629-1695) saxelebTan, Tumca calke-

uli amocanebi, romlebic dakavSirebuli iyo azartul TamaSebSi Sanse-

bis daTvlasTan, XV-XVI saukuneebSic ganixileboda italieli maTema-

tikosebis mier (kardano, paColi, tartalia da sxva). am mecnierebis

WeSmariti istoria ki iwyeba i. bernulis (1654-1705), p. laplasisa

(1749-1827) da s. puasonis (1781-1840) Sromebidan. maTematikuri stat-

istika rogorc mecniereba iwyeba k. gausis (1777-1855) Sromebidan. ma-Tematikuri statistikis ganviTarebaSi didi wvlili miuZRviT k. pirs-

ons (1857-1936), r. fiSers (1860-1962), e. neimans (1894-1977), a. kol-

mogorovs, n. smirnovs (1900-1966), a. valds (1902-1950) da sxvebs.

saqarTveloSi albaTobis Teoriisa da maTematikuri statistikis

swavlebis tradicias safuZveli daudo andria razmaZem (1889-1929),

romelic kiTxulobda leqciebis kurss axlad daarsebul Tbilisis

universitetSi, xolo maTematikis am dargis ganviTarebas safuZveli

Cauyara gvanji maniam (1918-1985), masve ekuTvnis pirveli qarTuli sa-

xelmZRvaneloebi da monografiebi am mimarTulebiT. aRsaniSnavia, rom

saqarTveloSi albaTobis Teoriisa da maTematikuri statistikis ganvi-Tarebas ara marto wminda maTematikuri gamokvlevebiT, aramed gamoyen-

ebiTi xasiaTis kvlevebiTac Rrma kvali daaCnia revaz CitaSvilma

(1942-1995).

maTematikuri statistika farTod gamoiyeneba teqnikur gamokvl-

evebSi, sicocxlis Semswavlel mecnierebebSi, ekonomikaSi, sabanko sfe-

roSi, marTvis (menejmentis) Teoriasa da praqtikaSi, sociologiaSi,

medicinaSi, geologiaSi, istoriaSi da a. S. dakvirvebebis, gazomvebis,

eqsperimentebis Sedegebis damuSavebasa da analizTan saqme gvaqvs rog-

orc adamianis praqtikuli moRvaweobis yvela sferoSi, ise samecniero

kvlevebis yvela mimarTulebaSi. statistikis aRwerilobiTi mxaridannaTeli xdeba, Tu adamianis moRvaweobis romel mxares exeba esa Tu is

amocana: ekonomikas, demografias, biologias, medicinas, sociologias

da a.S. Sesabamisad, SeiZleba visaubroT ekonometrikaze, demografiul

statistikaze, biometriaze, bio-samedicino statistikaze, statistika-

ze socialur mecnierebebSi da a.S., rogorc statistikis erTi Sexedv-

iT sxvadasxva sferoebze. sinamdvileSi, yoveli maTgani warmoadgens

monacemTa sxvadasxva bazas, romelTa damuSaveba da analizi xdeba maTe-

matikuri statistikis meTodebis gamoyenebiT.

7

silabusisaswavlo

kursis

dasaxeleba

albaToba da maTematikuri statistika qimiis,

biologiisa da sicocxlis Semswavleli mecnierebe-

bisTvis

saswavlo

kursis kodi

saswavlo

kursis

statusi

kursi gaTvaliswinebulia qimiis, biologiisa da sic-

ocxlis Semswavleli mecnierebebis meoTxe kursis

studentebisTvis, rogorc savaldebulo

saswavlo

kursis

xangrZlivoba

erTi semestri

ECTS-saswavlo

kursi

kreditebi

6 krediti: 60 sakontaqto saaTi (leqcia 30 saaTi,

praqtikuli 30 saaTi, laboratoriuli samuSao

(cxrilis gareT)) 90 saaTi damoukidebeli muSaobis-

aTvis

leqtorebi prof. omar furTuxia, Tsu zust da sabunebismetyve-

lo mecnierebaTa fakulteti, maTematikis mimarTule-

ba, telefoni: 304145 (samsaxuri), 189346 (bina),899503082 (mobiluri), e-mail: [email protected];[email protected]. zurab cigroSvili, saqarTvelos teqnikuri un-

iversiteti, qarTul-amerikuli universiteti, telef-

oni: 304145 (samsaxuri), 899317024 (mobiluri), e-mail: [email protected] manjgalaZe, fiz.-maT. mecnierebaTa kandidati,

mowveuli pedagogi, Tsu zust da sabunebismetyvelo

mecnierebaTa fakulteti, telefoni: 304145 (samsaxu-

ri), 968771 (bina), e-mail: [email protected]

kursis

mizani

leqciebis kursi gankuTvnilia iv. javaxiSvilis saxe-

lobis Tbilisis saxelmwifo universitetis qimiis,

biologiisa da sicocxlis Semswavlel mecnierebaTa

mimarTulebebis studentebisTvis. misi mizania gamoum-uSaos studentebs realur monacemebTan muSaobis un-

ar-Cvevebi.

saswavlo

kursis

Seswavlis

wina

pirobebi

umaRlesi maTematika, nawili I – kalkulusi

mailto:[email protected]




8

saswavlokursis

formati

leqcia, praqtikuli, praqtikuli samuSao

saswavlo

kursis

Sinaarsi

Sesavali. 1. statistikis sagani. populacia da SerCe-

va. aRweriTi da daskvniTi statistikis amocanebi. (ix.

[1], Tavi I.1; [4], Tavi 1)

2. monacemebis pirveladi damuSaveba. 2.1. monacemebis

tipebi. 2.2. monacemebis grafikuli warmodgena. (ix.

[1], Tavi I.2-I.3, I.6; [2], Tavi 2, $ 1-2; [4], Tavi 2)

2 sT leqcia, 2 sT praqtikuli

3. ricxviT monacemTa SerCeviTi maxasiaTeblebi. 3.1. sa-

Sualo yofaqcevis maxasiaTeblebi. 3.2. SerCevis gafant-

ulobis (ganfenilobis) maxasiaTeblebi. 3.3. SerCeviTiricxviTi maxasiaTeblebis Tvisebebi. 3.4. dawyvilebuli

monacemebi. korelacia. (ix. [1], Tavi I.7-I.9; [2], Tavi 3,

$ 1-3; [4], Tavi 3)


4. albaTobis Teoriis elementebi. 4.1. SemTxveviTi eq-

sperimenti, albaTuri sivrce, xdomiloba, SemTxveviTi

sidide. 4.2. moqmedebebi xdomilobebze. 4.3. xdomilo-

bis albaToba, albaTobaTa Tvisebebi. 4.4. pirobiTi

albaToba. albaTobaTa gamravlebis wesi. xdomilobaTa

damoukidebloba. 4.5. sruli albaTobisa da baiesisformulebi. (ix. [1], Tavi II.1-II.2; [2], Tavi 4, $ 1-5;

[3] Tavi I.1-I.2; [4], Tavi 5)


5. diskretuli SemTxveviTi sidideebi. ZiriTadi alba-

Turi ganawilebebi. 5.1. SemTxveviTi sidideebi da maTi

ricxviTi maxasiaTeblebi. 5.2. diskretuli SemTxvevi-

Ti sididis ganawilebis funqcia. 5.3. gadanacvlebebi

da jufdebebi. 5.4. binomuri ganawileba. 5.5. puasonis

ganawileba. 5.6. kavSiri binomur da puasonis ganawil-

ebebs Soris. (ix. [1], Tavi II.3-II.4; [2], Tavi 5, $ 1-6;[3] Tavi I.3-I.4; [4], Tavi 6)


6. uwyveti SemTxveviTi sidideebi. ZiriTadi albaTuri

ganawilebebi. 6.1. uwyveti SemTxveviTi sidideebi da

maTi ganawileba. 6.2. normaluri SemTxveviTi sididee-

bi (normaluri ganawileba). 6.3. binomuri da puasonis

ganawilebebis aproqsimacia normaluri ganawilebiT.

(ix. [1], Tavi II.5-II.6; [2], Tavi 6, $ 1-4; [3] Tavi I.5-

I.7; [4], Tavi 7)

9

2 sT leqcia, 2 sT praqtikuli7. SefasebaTa Teoria. wertilovani Sefasebebi. 7.1.

Sesavali. 7.2. wertilovani Sefasebebi. 7.3. central-

uri zRvariTi Teorema. 7.4. binomuri populaciis pparametris wertilovani Sefaseba. 7.5. puasonis popu-

laciis parametris wertilovani Sefaseba. (ix. [1],

Tavi III.3-III.4; [2], Tavi 7, $ 1-5; [3] Tavi II.1)


8. SefasebaTa Teoria. intervaluri Sefasebebi. 8.1. Se-savali. 8.2. populaciis saSualos intervaluri Sefa-

sebebi. 8.3. normaluri populaciis dispersiis inter-

valuri Sefaseba. 8.4. binomuri populaciis p paramet-ris intervaluri Sefaseba. 8.5. puasonis ganawilebis

parametris intervaluri Sefaseba. (ix. [1], Tavi

III.5; [2], Tavi 8, $ 1-4; [3] Tavi II.1-II.3; [4], Tavi 8)

2 sT leqcia, 2 sT praqtikuli9. hipoTezaTa Semowmeba. erTamokrefiani amocanebi.

9.1. Sesavali, ZiriTadi cnebebi. 9.2. hipoTezis Semow-

meba normaluri populaciis saSualosaTvis cnobili

dispersiis dros. 9.3. hipoTezis Semowmeba normalu-

ri populaciis saSualosaTvis ucnobi dispersiis

dros. 9.4. kriteriumis simZlavris gamoTvla. 9.5. Se-

rCevis minimaluri moculobis gansazRvra. 9.6. hipoT-

ezis Semowmeba normaluri populaciis dispersiisaTv-

is (ormxrivi alternativa). 9.7. hipoTezis Semowmeba

binomuri populaciis p parametris Sesaxeb (ormxrivialternativa). 9.8. hipoTezis Semowmeba puasonis pop-

ulaciis parametris Sesaxeb (mcire moculobis Se-

rCevebisaTvis). (ix. [1], Tavi III.6; [2], Tavi 9, $ 1-8;

[3] Tavi II.4-II.8; [4], Tavi 9)


10. hipoTezaTa Semowmeba. oramokrefiani amocanebi.

10.1. Sesavali. 10.2. dawyvilebuli monacemebi. 10.3.

oramokrefiani t -kriteriumi toli, ucnobi disper-siebis SemTxvevaSi. 10.4. hipoTeza ori normaluri po-

pulaciis dispersiaTa tolobis Sesaxeb. 10.5. oramo-

krefiani t -kriteriumi aratoli dispersiebis Sem-

TxvevaSi. 10.6. SerCevaTa moculobebis gansazRvra.

ori populaciis saSualoebis Sedarebis kriteriumis

simZlavre. (ix. [1], Tavi III.7; [2], Tavi 10, $ 1-6; [3]

Tavi II.9-II.10; [4], Tavi 10)


10

11. mravalamokrefiani amocanebi. dispersiuli anali-zi. 11.1. Sesavali. 11.2. hipoTezaTa Semowmeba erTfaqt-

orian ANOVA modelSi. deterministuli efeqtebis

SemTxveva. 11.3. jgufTa Sedareba erTfaqtorian ANO-VA modelSi. wyvilTa Sedarebis t -kriteriumi. 11.4.

wrfivi kontrastebi. 11.5. mravlobiTi Sedareba (bonf-

eronisa da Sefes meTodebi). 11.6. hipoTezaTa Semowme-

ba erTfaqtorian ANOVA modelSi SemTxveviTi efeq-

tebis dros. (ix. [1], Tavi III.8; [2], Tavi 11, $ 1-6; [3]

Tavi II.19; [4], Tavi 12)

2 sT leqcia, 2 sT praqtikuli12. oramokrafiani amocanebi binomuri proporciebisa-

Tvis. kategoruli monacemebi. 12.1. Sesavali. 12.2 ori

binomuri proporciis Sedareba, normaluri aproqsim-

acia. 12.3 ori binomuri proporciis Sedareba, SeuRl-

ebis 22 cxrili. 12.4. SeuRlebis 22 cxrili. niSa-

nTa damoukidebloba. (ix. [1], Tavi III.9; [2], Tavi 12, $

1-2; [3] Tavi II.11; [4], Tavi 10)

2 sT leqcia, 2 sT praqtikuli12.5. kategoruli monacemebis efeqtebis sazomebi.

12.6. fiSeris zusti kriteriumi. 12.7. maknemaris kri-

teriumi proporciebisaTvis dawyvilebul monacemebSi.

(ix. [1], Tavi III.9; [2], Tavi 12, $ 3-5; [3] Tavi II.9-

II.11; [4], Tavi 10)


12.8. SerCevis moculobis gansazRvra da kriteriumis

simZlavre ori binomuri proporciebis Sedarebisas.

12.9. SeuRlebis r c cxrilebi. 12.10. Tanxmobis

2 kriteriumi. (ix. [1], Tavi III.10-III.11; [2], Tavi12, $ 6-8; [3] Tavi II.14-II.16; [4], Tavi 12)


13. regresiuli analizi da korelacia. 13.1. Sesavali,

ZiriTadi cnebebi. 13.2. umcires kvadratTa meTodi.

13.3. regresiis wrfis parametrebis aRricxva. 13.4.Tanxmobis kriteriumebi regresiis wrfisaTvis. 13.5.

intervaluri Sefasebebi wrfivi regresiisaTvis. 13.6.

naSTTa analizi martivi wrfivi regresiisaTvis. 13.7.

korelaciis koeficienti. 3.8. kerZo da mravlobiTi

korelacia. 13.9. mravlobiTi logisturi regresia.

(ix. [1], Tavi III.13; [2], Tavi 13, $ 1-11; [3] Tavi II.17-

II.18; [4], Tavi 11)


11

14. daskvniTi statistikis araparametruli meTodebi.14.1. Sesavali. 14.2. niSnebis kriteriumi. 14.3. uilko-

ksonis niSniani rangebis kriteriumi. 14.4. uilkoks-

onis rangTa jamis kriteriumi. 14.5. Tanadoba uilko-

ksonis rangTa jamis kriteriumsa da 2 -kriteriums

Soris. 14.6. kraskel-uolisis kriteriumi. 14.7. spir-

menis rangobrivi korelaciis koeficienti. (ix. [1],

Tavi III.12; [2], Tavi 14, $ 1-7; [3] Tavi II.18)

2 sT leqcia, 2 sT praqtikuliliteratura

1. o.furTuxia. aRweriTi statistika, albaToba, sta-

tistikuri daskvnebis Teoria. Tbilisi, 2008.

2. B. Rosner. Fundamentals of Biostatistics. Published byDuxbury, 1995.3. o. furTuxia. albaToba da statistika magaliTebsa

da amocanebSi. Tbilisi, 2009.

4. Allan G. Bluman. Ementary Statistics: a brief version,second edition. Published by McGraw-Hill, New York,2003.

Sefaseba kolokviumi (weriTi formiT, sami sakiTxi, TiToeuli

swori pasuxi fasdeba 5 qulamde);

saboloo gamocda oTxsakiTxiani bileTebiT. TiTeul

sakiTxze pasuxi fasdeba 10 qulamde).1. daswreba 10%2. praqtikuli mecadineoba (15%),

laboratoriuli samuSao (5%)

20%

3. kolokviumi 15%4. kolokviumi 15%5. saboloo gamocda 40%

saboloo Sefaseba 100%

* gamocdaze daSvebis winapiroba: aranakleb 30

qulisa 1-4 komponentebSi

* kreditis miniWebis aucilebebi piroba: aranakleb

21 qulisa saboloo gamocdaSi

savaldebulo

literatura

1. o.furTuxia. aRweriTi statistika, albaToba, sta-

tistikuri daskvnebis Teoria. Tbilisi, 2008.

2. B. Rosner. Fundamentals of Biostatistics. Published byDuxbury, 1995.3. o. furTuxia. albaToba da statistika magaliTebsa

da amocanebSi. Tbilisi, 2009.4. Allan G. Bluman. Ementary Statistics: a brief version,

12

second edition. Published by McGraw-Hill, New York,2003.

damatebiTi

literatura

da sxvasaswavlo

masala

1. g. mania. albaTobis Teoria da maTematikuri

statistika. Tsu, Tbilisi, 1976

2. n. lazrieva, m. mania, g. mari, a. mosiZe, a. toronj-aZe, T. toronjaZe, T. ServaSiZe. albaTobis Teoria da

maTematikuri statistika ekonomistebisaTvis. fondi

«evrazia», Tbilisi, 2000.

3. o. furTuxia. albaTobis Teoria da maTematikuri

statistika. Tbilisi, 2007.4. P. Newbold, W. L. Carlson, B. M. Thorne. Statistics forBusiness and Economics, sixth edition. Prentice Hall, UpperSaddle River, New Jersey, 2007.5. В. Феллер. ВВедение в теорию вероятностей и ееприложения. Москва, 1967.

swavlis

Sedegi

miznidan gamomdinare, swavlis Sedegi unda iyos is,

rom studentebs SeeZloT statistikur meTodebze

dayrdnobiT damoukideblad daamuSaon maT mier mopov-

ebuli realuri monacemebi da gaakeTon swori daskvn-

ebi Sesaswavli movlenis Sesaxeb, rasac cxadia, gamo-

iyeneben TavianT pirdapir saqmianobaSi.

13

I. aRweriTi statistika

l e q c i a 1.

Tavi 1. statistikis sagani. populacia da SerCeva.aRweriTi da daskvniTi statistikis amocanebi.

Cven xSirad gvesmis Semdegi tipis gancxadebebi da mtkicebuleb-

ebi: “dou-jonsis saSualo daeca 6 punqtiT”, “samomxmareblo saqonel-

ze fasebis indeqsi gasuli Tvis ganmavlobaSi gaizarda 8%-iT”, “ukana-

skneli gamokiTxva gviCvenebs, rom prezidentis reitingi amJamad Seadg-

ens 63%-s”, “pacientTa 98%-s, romelic imyofeboda klinikur gamokv-

levaze, ar ganucdia raime gverdiTi efeqti mkerdis kibos axali prep-

aratisagan”. Cven sworad unda gavaanalizoT da mivceT Sesabamisi int-

erpretacia yovel monacems. xSiria SemTxvevebi, roca imisaTvis rom

mimdinare movlenebs mieces gonivruli (inteleqtualuri) Sefaseba,Cven gvesaWiroeba (gvixdeba) gadavamuSaoT monacemebis arsebiTad didi

raodenoba. mTavroba da mecnier-mkvlevarebi xarjaven milinobiT dol-

ars, raTa moagrovon monacemebi. federaluri mTavrobebi xels uwyoben

monacemebis Segrovebas, rogorc sakuTari ZalisxmeviT, ise korporaci-

ebis mimarT moTxovniT – raTa maT misawvdomi gaxadon informaciebi.

gadawyvetilebebi xSirad dafuZnebulia arasrul informaciaze.

magaliTad, sagamocdo komisiis mier universitetSi miRebuli pirveli

kursis studentebi ise irCeven TavianT ZiriTad specialobas, rom ara

aqvT naTeli warmodgena momavali karieris Sesaxeb. an pacientebi, rom-

lebic daavadebuli arian kiboTi, Tanxmdebian monawileoba miiRon axa-

li preparatis klinikur gamokvlevaze ise, rom maTTvis ucnobia ampreparatis moqmedebis yvela SesaZlo Sedegi. analogiurad, is adamian-

ebi, romlebic regularulad Rebuloben saqmian gadawyvetilebebs gar-

emo pirobebis gaTvaliswinebiT, ar SeiZleba darwmunebulebi iyvnen im

faqtorebis momaval yofaqcevaze, romlebic saboloo gavlenas axdenen

gansaxilveli movlenebis saboloo Sedegze.

mewarmis mier SemoTavazebuli kontraqtis safasuri ar SeiZle-

ba iyos srulad garkveuli, is ver moicavs mosalodnel fasebs momav-

alSi da ver iqneba masSi gaTvaliswinebuli konkurentebis mier SemoT-

avazebuli fasebi. am dasanani gaurkvevlobis miuxedavad, mewarmem unda

daawesos Tavisi fasi. investorma garantirebulad ar icis finansuribazari iqneba aRmavali, gawonasworebuli Tu daRmavali. miuxedavad am-

isa, investorma unda gadawyvitos rogor daabalansos Tavis finansuri

portfeli aqciebiT, obligaciebiT da finansuri bazris sxva instrum-

14

entebiT, maSin roca finasuri bazris momavali ganviTareba misTvis uc-nobia.

aRsaniSnavia, rom adamianebis umravlesobisTvis albaToba da

statistika axlobeli gaxda radios, televiziis, gazeTebisa da Jurn-

analebis saSualebiT. magaliTad, gazeTebSi napovni iyo Semdegi gancxa-

debebi: a). Cveulebriv, dReSi erTi abi vitamini da minerali asakovan

adamianebSi zrdis garkveul imunur reaqciebs 64 procentiT, im kvle-

vebis Tanaxmad, romelic Catarebuli iqna niu jersis medicinisa da

stomatologiis universitetSi (USA Weekend, January 6-8, 1995); b).qveynis masStabiT gamokiTxuli 1 000 ojaxidan, 40% ganacxada, rom

floben sul cota erT ukabelo telefons, 9% flobs ors an mets(Tribune-Rev-iew, Greensburg, PA, Jenuary 8, 1995, p.B3); g). 1994 wels

sacalo vaWrobis specialurma saTxilamuro sportis maRaziebma gayid-

es 760 000 Txilamuri saSualo fasad 153$ da 176 200 saTxilamu-

ro kostumi saSualo fasad 280$ (Tribune-Review, Greensburg, PA, Jenu-ary 8, 1995, p. G1); d). dortmondSi (pensilvania) gayiduli uZravi qone-

bis saSualo fasi bo-lo 12 Tvis ganmavlobaSi iyo 64 304$ (Tribune-Review, Greensburg, PA, Jenuary 8, 1995, p.B3); e). aSS-is moqalaqeebi

wlis ganmavlobaSi sportul tansacmelze saSualod xarjaven 193$(USA Today, Jenuary 10, 1995).

statistika gamoiyeneba adamiani moRvaweobis TiTqmis yvela sfe-

roSi. magaliTad, sportSi statistikoss SeuZlia awarmoos aRricxva

iardebis ricxvisa rac gairbines fexburTelebma fexburTis matCisganmavlobaSi an daiTvalos raodenoba dartymebisa rac beisbolistma

gaakeTa sezonis ganmavlobaSi. sxva sferoebSi, rogoricaa jandacva,

administratori dainteresebulia im moqalaqeebis raodenobiT, romle-

bic avadmyofobdnen axali Stamis virusiT garkveuli periodis ganmav-

lobaSi. ganaTlebaSi, mkvlevars SeiZleba ainteresebdes swavlebis axa-

li meTodi ukeTesia Tu ara, vidre Zveli. aq CamoTvlilia mxolod

ramodenime magaliTi, Tu rogor SeiZleba iqnes gamoyenebuli statist-

ika sxvadasxva mimarTulebebiT. garda amisa, statistika gamoiyeneba ga-

mokvlevis Sedegebis gasaanalizeblad da rogorc instrumenti samecni-

ero kvlevebSi gadawyvetilebis misaRebad, romelic dafuZnebulia eqs-perimentebze.

monacemebi SeiZleba gamoyenebul iqnes sxvadasxvanairad. codnis

is sistema, romelsac statsitikas uwodeben, imis mixedviT Tu rogor

gamoiyeneba monacemebi, zogjer iyofa or ZiriTad mimarTulebad. es

ori mimarTulebaa: 1. aRweriTi statistika da 2. statistikuri daskv-nebi.

aRweriT statistikaSi statistikosi cdilobs aRweros situa-

cia. ganvixiloT mosaxleobis nacionaluri aRweris sakiTxi, romelsac

aSS mTavroba atarebs yovel aT weliwadSi. am aRweris Sedegebi gvaZ-

15

levs mosaxleobis saSualo asaks, Semosavals da sxva gansakuTrebulmaxasiaTeblebs aSS-is mosaxleobis. am informaciis misaRebad, mosaxl-

eobis aRweris biuros unda gaaCndes garkveuli saSualebebi Sesabamisi

monacemebis Sesagroveblad. mas Semdeg rac miRebulia (Segrovebulia)

monacemebi, aRweris biurom unda moaxdinos maTi organizeba da dajam-

eba. sabolood, aRweris biuros esaWiroeba saSualebebi (meTodebi) ra-

Ta miRebuli monacemebi warmoadginos azriani (an TvalsaCino) formiT,

magaliTad, cxrilebis, grafikebis an diagramebis formiT.

istoriuli SeniSvna. aRweriTi statistikis sawyisebi saTaves

iRebs monacemTa Segrovebis meTodebiT, romlebic gamoiyeneboda mosax-

leobis aRwerisas babilonsa da egvipteSi Zv. w. aRricxvis 4500 --3000 wlebSi. garda amisa, romis imperatorma augustma (Zv. w. aRricx-

vis 27--17 wlebi) Caatara gamokvleva romis imperiis moqalaqeebis ro-

gorc dabadebisa da sikvdilianobis Sesaxeb, ise yoveli moqalaqis ku-

TvnilebaSi myofi saqonlis raodenobisa da maT mier wlis ganmavloba-

Si moyvanili sasoflo-sameurneo kulturebis Sesaxeb.

aRweriTi statistika gulisxmobs monacemebis Segrovebas, orga-

nizebas, dajamebasa da warmodgenas.

statistikis meore ganStoebas uwodeben statistikuri daskvne-bis Teorias. aq, statistikosi cdilobs gaakeTos daskvnebi populaci-is SerCevebidan (amorCevebidan). statistikuri daskvnebi iyenebs albaT-obas, e. i. xdomilebis moxdenis Sanss. adamianebis didi nawilisaTvis

cnobilia albaTobis principialuri sqemebi azartuli TamaSebis sxva-

dasxva formebidan. adamiani romelic TamaSobs karts, saTamaSo kamaT-

els, bingosa da latareas, igebs an kargavs (agebs) albaTuri ganawil-

ebis kanonis mixedviT. albaTobis Teoria aseve gamoiyeneba sadazRvevo

saqmianobaSi da adamianis moRvaweobis sxva mraval sferoSi.

istoriuli SeniSvna. statistikuri daskvnebis Teoria saTaves

iRebs 1600 wlidan, rodesac jon grantma gamoaqveyna wigni mosaxleo-

bis zrdis Sesaxeb, sadac sikvdilianobis cxrilze dayrdnobiT gakeTe-

bulia bunebrivi da politikuri xasiaTis SeniSvnebi. TiTqmis imavedros, maTematikosma da astronomma edmund halim gamoaqveyna sikvdil-

ianobis pirveli srulyofili cxrili (aRsaniSnavia, rom sadazRvevo

kompania iyenebs sikvdilianobis cxrilebs dazRvevis tarifis dasadge-

nad).

SemoviRoT axla populaciisa da SerCevis cnebebi da vnaxoT ra

gansxvavebaa maT Soris.

populacia Sedgeba yvela im obieqtisagan, romelic Seiswavleba

(is aris dakvirvebis yvela SesaZlo SedegTa simravle). umetes SemTx-

vevaSi, rigi mizezebis gamo (magaliTad, danaxarjebis siZvire, drois

simcire, populaciis moculobis sidide, samedicino problemebi da a.S.) mkvlevars ara aqvs SesaZlebloba statistikuri kvlevisaTvis gam-

16

oiyenos mTliani populacia. amitom mkvlevari, rogorc wesi, iyenebsSerCevas.

SerCeva aris obieqtebis garkveuli jgufi (nawili) amorCeuli

populaciidan. Tu SerCevis kiTxvebi (obieqtebi) sworadaa SerCeuli,

maSin umetes SemTxvevaSi isini iqnebian igive an analogiuri im maxasi-

aTeblebis rac kiTxvebs (obieqtebs) gaaCniaT populaciaSi.

populaciis srul aRweras Cven SerCevis gamoyofas da mis Sewa-

vlas vamjobinebT Tundac imitom, rom nawilis dakvirveba ufro iafia,

vidre mTelisa, Tumca xarjebis ekonomiis garda SerCevis upiratesob-

as sxva motivebic ganapirobeben. SerCevis umniSvnelovanesi saxeobaa

SemTxveviTi SerCeva. SemTxveviTi SerCeva gulisxmobs, rom populaciisyoveli elementisaTvis gansazRrulia SerCevaSi moxvedris Sansi, alb-

aToba. rodesac SerCevis gegma iseTia, rom populaciis yoveli elemen-

tisaTvis SerCevaSi moxvedris Sansi erTnairia, amboben, rom gvaqvs ma-rtivi SerCeva; SerCeva SemTxveviTi ricxvebis gamoyenebiT; e. w. siste-maturi SerCeva (vTqvaT, yoveli me-10 erTeuli); ganSrevebuli SemTxv-eviTi SerCeva – uzrunvelyofs SerCevaSi popu-laciis sxvadasxva jgu-

fis warmomadgenlobas; klasteruli SerCeva – populacias vyofT Tan-

aukveT nawilebad – klasterebad, SemTxve-viT virCevT klasters da

Semdeg SerCeuls srulad aRvwerT.

statistikuri daskvnebis Teoriis nawils, romelic warmoadg-ens populaciaze winadadebis Sefasebisas gadawyvetilebis miRebis

process, dafuZnebuls SerCevidan miRebul informaciaze, hipoTezaTaSemowmeba ewodeba. magaliTad, mkvlevars SeiZleba survili hqondes

icodes amcirebs Tu ara axali preparati gulis SetevaTa raodenobas

70 welze ufro didi asakis mamakacebSi. amis gamosakvlevad, arCeuli

iqneba 70 wels gadacilebul mamakacTa ori SerCeva (jgufi). erT-erTi

SerCevis mamakacebs eZlevaT aRniSnuli preparati, xolo meore SerCev-

aSi myof mamakacebs aZleven raRac nivTierebas, romelsac janmrTelob-

is TvalsazrisiT adamianisaTvis arc sargeblobis motana SeuZlia da

arc zianis (e. w. placebo). eqsperimentis bolos iTvlian mamakacebisTiToeul jgufSi momxdar gulis SetevaTa raodenobas, Semdeg atareb-

en statistikuri hipoTezis Semowmebis process da Rebuloben gadawy-

vetilebas preparatis efeqturobis Sesaxeb.

statistikosebi statistikur monacemebs iyeneben agreTve imis

gasarkvevad, Tu ra kavSiria sidideebs Soris? magaliTad, ukanaskneli

aTwleulebis ganmavlobaSi gansakuTrebuli yuradReba eqceva kavSiris

dadgenas mowevasa da filtvebis kibos Soris. 1964 wels aSS-is mTav-

arma qirurgma gamoaqveyna naSromi “moweva da janmrTeloba”, sadac na-

Tqvamia, rom monacemebis ganxilvisa da Sefasebis Semdeg misma jgufma

aRmoaCina garkveuli kavSiri mowevasa da filtvebis kibos Soris. masar uTqvams, rom sigaretis moweva aris filtvebis kibos faqtiuri mi-

17

zezi, magram ambobs, rom maT Soris aris kavSiri. es daskvna dafuZneb-uli iyo hamondisa da hornis mier 1958 wels Catarebul gamokvleva-

ze. am kvlevisas dakvirvebul iqna 187783 mamakaci 45 Tvis ganmavloba-

Si. aRmoCnda, rom filtvebis kiboTi gamowveuli sikvdilianobis koef-

icienti mamakacebis am jgufSi 10-jer ufro didi iyo mwevelebSi ara-

mwevelebTan SedarebiT.

dabolos, warsuli da mimdinare monacemebisa da pirobebis Ses-

wavliT statistikosebi cdiloben am informaciis safuZvelze gaakeT-

on prognozi (iwinaswarmetyvelon ra iqneba momavalSi). magaliTad, av-

tomobilebis dilers (gamyidvels), romelic naxavs gasuli gayidvebis

Canawerebs konkretuli Tveebis mixedviT, SeuZlia gadawyvitos Tu ratipis da ramdeni avtomobili (ama Tu im tipis) dasWirdeba momavali

wlis konkretul TveSi.

statistikuri daskvnebis Teoria moicavs ganzogadoebas SerCev-

idan populaciamde, hipoTezaTa Semowmebis kriteriumebis dadgenas, si-

dideebs Soris kavSiris dadgenas da prognozis gakeTebas.

sabolood, Cven mivdivarT Semdeg ganmartebamde.

statistika aris mecniereba, romelic erTis mxriv Seiswavlis

monacemebis Segrovebis, organizebis, klasifikaciis, sisitematizaciisa

da pirveladi damuSavebis meTodebs, xolo meores mxriv, Seiswavlis

damuSavebuli monacemebis analizisa da gamoyenebis meTodebs Teoriu-li da praqtikuli daskvnebis miRebis mizniT. Sesabamisad, maTematiku-

ri statistika iyofa or nawilad: aRweriTi (e. w. deskrifciuli)

statistika da statistikuri daskvnebis Teoria, Tu ar CavTvliT al-

baTobis Teorias, romelic warmoadgens xids statistikis am or naw-

ils Soris. ufro zustad albaTobis Teoria warmoadgens statistik-

uri daskvnebis Teoriis maTematikur safuZvels.

statistikis aRwerilobiTi mxaridan naTeli xdeba, Tu moRvawe-

obis romel sferos exeba esa Tu is amocana, ekonomikas, demografias,

biologiasa Tu medicinas. amdenad, SeiZleba saubari ekonometrikaze,

demografiul statistikaze, biometriasa Tu biosamedicino statisti-kaze, rogorc statistikis erTi SexedviT sxvadasxva sferoebze. sinam-

dvileSi, yoveli maTgani warmoadgens monacemTa sxvadasxva bazas, rom-

elTa damuSaveba da analizi xdeba maTematikuri statistikis meTodeb-

is gamoyenebiT. leqciaTa qvemoTmoyvanili kursiss mizania maTematiku-

ri statistikis meTodebis rolis warmoCineba da maTi gamoyeneba eko-

logiuri, biologiuri da samedicino praqtikidan mosuli monacemebis

analizisaTvis. moviyvanoT mokle CamonaTvali im eko-biologiuri da

samedicino sferoebisa, romelTa amocanebsac rogorc magaliTebs Cven

gamoviyenebT Cvens kursSi. es aris:

baqteriologia (Bacteriology), botanika (Botany), simsivnuri da-avadebebi (Cancer), kardiologia (Cardiology), gulsisxlZarRvTa daava-

18

debebi (Cardiovascular Disease), tvinis daavadebebi (Cerebrovascular Dise-ase), demografia (Demography), kanis daavadebebi (Dermatology), diab-eti (Diabets), endokrinologia (Endocrinology), garemos dacva (Enviro-nmental Health), epidemiologia (Epidemiology), kuWis daavadebebi (Gast-roenterology), genetika (Genetics), ginekologia (Gynecology), hematol-

ogia (Hematology), RviZlis daavadebebi (Hepatic Disease), infeqciuridaavadebebi (Infectious Disease), fsiqikuri daavadebebi (Mental Health)mikrobiologia (Microbiology), ofTalmologia (Opthalmology), otola-

ringologia (Otolaringology), pediatria (Pediatrics), farmakologia

(Pharmacology), revmatologia (Rheumatology), sasporto medicina

(Sports Medicine) da a.S.imisaTvis, rom warmodgena SeviqmnaT imaze, Tu risi Seswavla

SeuZlia statistikas, anu ra tipis amocanebis dasma da gadawyveta Se-

uZlia maTematikuri statistikis meTodebs, moviyvanoT aseTi magaliTi

samedicino (epidemiologiis) praqtikidan:

magaliTi 1.1. XX saukunis 40-50 wlebSi aSS-Si sazogadoebrio-

bis yuradReba mipyrobili iyo poliomieliTis epidemiisadmi. am serio-

zuli problemis gadaWris mizniT jonas selkma (Jonas Salk) pitsburg-is universitetidan Seqmna vaqcina, romlis efeqturobac daadastura

ramodenime winaswarma laboratoriulma gamokvlevam. vaqcinis sazogad-

od efeqturobis dasadgenad aucilebeli gaxda masobrivi kvlevis Cat-areba. magram amisaTvis saWiro iyo eqsperimentis sworad dagegmva da

misi sworad warmarTva. SemTxveviTi SerCevis meTodis (ix. qvemoT) sa-

fuZvelze SerCeul iqna ori milioni amerikeli skolis moswavle, ro-

melTa naxevars gaukeTes selkis vaqcina, xolo meore naxevars –

placebo (Placebo) (uvnebeli wamali – matyuara). kvlevis centrma da-

afiqsira, Tu vis gaukeTda selkis vaqcina da vis – placebo. maTemati-

kuri statistikis meTodebis gamoyenebiT daasabuTes, rom poliomieli-

TiT inficirebulTa raodenoba gacilebiT dabalia selkis vaqciniT

acril moswavleebs Soris, vidre maT Soris visac gaukeTda placebo.

aqedan daaskvnes, rom selkis vaqcina efeqturia sazogadod poliomie-liTis winaaRmdeg da is SemoiRes sayovelTao xmarebaSi.

upirveles yovlisa, SevniSnoT, rom statistikosisaTvis, rome-

lsac surda Seefasebina vaqcinis efeqturoba saskolo asakis bavSvebis

mTeli populaciisaTvis, xelmisawvdomi iyo am populaciis mxolod

nawilis – SemTxveviT SerCeuli ori milioni acrili bavSvis monacemi.

statistikaSi mTeli populaciis aseT nawils SerCevas, an amokrefasuwodeben. maSasadame, statistikosma daskvna unda gaakeTos garkveuli

ganusazRvrelobis pirobebSi da Sesabamisad, daskvniTi statistikis

ZiriTadi amocanaa SerCevidan miRebuli informaciis safuZvelze daskv-

nebis gakeTeba mTeli populaciisaTvis.

19

statistikuri amocanebis dasasmelad da gadasawyvetad mniSvne-lovania mTelis ra nawilzea saubari SerCevis (amokrefis) formirebi-

sas. magaliTad, ar SeiZleba daskvnebis gakeTeba adamianis simaRlis Se-

saxeb romelime populaciaSi, masSi mxolod kalaTburTelTa simaRle-

ebze dakvirvebebis safuZvelze. arsebiTia, rom amokrefaSi moxvedris

Sansi Tanabari unda hqondes kalaTburTelsac da Cia kacsac. aseT Sem-

TxvevaSi amboben rom, amokrefa unda iyos warmomadgenlobiTi anu rep-rezentatuli. swored es igulisxmeboda SemTxveviTi SerCevis meTodisqveS eqsperimentis dagegmvisas 1.1 magaliTSi. mxolod aseTi SerCevebis

safuZvelze gakeTebuli ganzogadebebia misaRebi statistikuri daskvne-

bisaTvis.statistikis Seswavlas Cven daviwyebT aRweriTi statistikis

amocanebiT, sadac SeviswavliT ricxviTi monacemebis warmodgenisa da

damuSavebis meTodebs. Semdeg gadavalT daskvniTi statistikis Seswav-

laze.

Tavi 2. monacemebis pirveladi damuSaveba.

2.1. monacemebis tipebi.

2.1.1. ricxviTi monacemebi:

diskretuli monacemebi gvxvdeba raime movlenis moxdenaTa rao-denobebze dakvirvebisas. magaliTad, saswrafo samedicino daxmarebis

sadgurSi erT saaTis ganmavlobaSi Semosul gamoZaxebaTa raodenoba;

petris lambaqze baqteriaTa koloniebis raodenoba da a.S.

uwyveti tipis monacemebi miiReba im sididis gazomvebis Sedeg-ad, romelsac SeuZlia garkveuli ricxviTi intervalis yvela mniSvne-

lobis miReba. magaliTad, wona, sigrZe, dro, temperatura, sisxlis

wneva da a.S.

2.1.2. kategoruli monacemebi:

atributuli monacemebi esaa Semdegi tipis ormniSvnelobebiani(diqotomiuri) monacemebi: ki/ara, qali/aci, avadmyofi/janmrTeli da

sxva.

nominaluri tipis monacemebs aqvT ramodenime daulagebeli kat-

egoriis mniSvneloba. magaliTad, avadmyofobis tipebi, mcenareTa saxeo-

bebi da a.S.

ordinaluri tipis monacemebs gaaCniaT ramodenime dalagebuli

kategoriis mniSvneloba. magaliTad, avadmyofobis stadiebi.

2.2. monacemebis grafikuli warmodgena.

2.2.1. variaciuli mwkrivi. empiriuli ganawilebis funqcia.magaliTi 2.1. cxrilSi mocemulia erT-erTi samSobiaro saxlis

20

axalSobilTa wonebi (gramebSi):

i xi i xi i xi i xi

1 3265 6 3323 11 2581 16 2759

2 3260 7 3649 12 2841 17 3248

3 3245 8 3200 13 3609 18 3314

4 3484 9 3031 14 2838 19 3101

5 4146 10 2069 15 3541 20 2834

rogorc vxedavT, mocemulia n = 20 moculobis SerCeva

3265, 3260, 3245, 3484, 4146, 3323, 3649, 3200, 3031, 2069,

2581, 2841, 3609, 2838, 3541, 2759, 3248, 3314, 3101, 2834,

romlebic Sesabamisad aRniSnulia x1, x2,..., xn simboloebiT. imisaTvis,

rom warmodgena SeviqmnaT am ricxvebis cvalebadobis sazRvrebis Sesa-

xeb, pirvel rigSi davalagoT es ricxvebi zrdadobis mixedviT, anu

statistikuri terminologiiT, SevadginoT monacemebis e.w. variaciulimwkrivi:

2069, 2581, 2759, 2834, 2838, 2841, 3031, 3101, 3200, 3245,

3248, 3260, 3265, 3314, 3323, 3484, 3541, 3609, 3649, 4146.

variaciuli mwkrivis elementebs sazogadod, Cven aRvniSnavT x(1)

x(2) ... x(n) simboloebiT. aq, sididiT mesame elements warmoadgens

rigiT me-16 monacemi, anu x(3) = x16 = 2759. sainteresoa davsvaT Sebru-nebuli kiTxva: sididiT meramdenea magaliTad, pirveli monacemi x1 =3265? variaciuli mwkrividan Cans, rom 3265 aris sididiT me-13 elem-

enti anu x1 = 3265 = x(13). am SemTxvevaSi amboben, rom pirveli monace-

mis rangia 13 da weren r1 =13. analogiurad ganisazRvreba meore, mesa-

me da a.S. monacemebis rangebic. ufro zustad, ix monacemis rangi

(rank) ewodeba am monacemis nomers variaciul mwkrivSi, Tu arcerTi

monacemi ar meordeba (sxva sityvebiT, ix -s rangi ewodeba iseT mTel

ricxvs ir , romlisTvisac sruldeba toloba: ( )ir ix x ). toli monacem-

ebis SemTxvevaSi ki TiToeuls unda mivaniWoT variaciul mwkrivSi ma-

Tze mosuli nomrebis saSualo ariTmetikuli.

gamovsaxoT variaciuli mwkrivis elementebi grafikulad. amisa-

Tvis yovel x wertilSi daviTvaloT variaciul mwkrivSi x-is marcxn-iv moTavsebuli elementTa raodenoba da SevafardoT is monacemTa sa-

erTo raodenobasTan, anu n-Tan. miRebul funqcias mocemuli SerCevis

Sesabamisi empiriuli ganawilebis funqcia ewodeba. mas Fn(x) simbolo-

Ti aRniSnaven:

Fn(x) = (1/n)#{im x(i), romelTaTvisac x(i) x}, (2.1)aq #{A} simboloTi aRniSnulia A simravlis elementebis raodenoba.

Cveni magaliTisaTvis empiriuli ganawilebis funqcias aqvs Semd-

egi saxe:

21

2.2.2. sixSiruli ganawileba.

magaliTi 2.2. qvemoT moyvanilia erT-erTi dabis SemTxveviTarCeul 87 ojaxSi skolis asakis bavSvTa raodenobebi:

ojaxSi bavSvebis

raodenoba

0 1 2 3 4 5 6 7

ojaxebis raodenoba 15 20 23 12 7 5 3 2 sul 87

am cxrilis pirvel striqonSi mocemulia ojaxebSi bavSvTa ra-odenobebis sxvadasxva SesaZlo variantTa mniSvnelobebi, xolo meore

striqonSi ki Sesabamisi sixSireebi. sxva msgavs monacemebTan Sesadareb-lad (magaliTad, igive sidides akvirdebodnen pirvel 16 ojaxSi) uf-

ro moxerxebulia es raodenobebi gadaviyvanoT e.w. fardobiT sixSire-ebSi an procentebSi, rogorc es gakeTebulia qvemoT moyvanil cxril-

Si:

ojaxSi bavSvebis

raodenoba

ojaxebis fardobiTi

sixSire

ojaxebi procentebSi

0 15/87 0.1724 17.24%1 20/87 0.2299 22.99%2 23/87 0.2644 26.44%3 12/87 0.1379 13.79%4 7/87 0.0805 8.05%5 5/87 0.0575 5.75%6 3/87 0.0345 3.45%7 2/87 0.0229 2.29%

jami 1 jami 100%

22

TvalsaCinoebisaTvis ukeTesia Sesabamisi sixSiruli cxrilebisSemdegi grafikuli warmodgena:

2.2.3. uwyvet monacemTa dajgufeba, histograma, poligoni.

uwyvet monacemTa didi raodenobebis warmosadgenad iyeneben mon-

acemTa kategoriebad dayofis anu monacemTa dajgufebis xerxs. sxva

sityvebiT rom vTqvaT, axdenen uwyveti monacemebis diskretizacias,anu maT ajgufeben kategoriebad (klasebad) da saubroben ara uwyvetimonacemis konkretul mniSvnelobaze, aramed im klasze, romelSic es

monacemi moxvda, rogorc diskretuli tipis monacemis erT-erT SesaZ-

lo mniSvnelobaze da amuSaveben aseT monacemebs, rogorc diskretuli

tipis monacemebs.

magaliTi 2.3. cxrilSi mocemulia sisxlis wnevis gazomvis Sed-

egebi (mm vwy. sv.) 120 avadmyofisaTvis:104.1 94.2 114.2 103.8 144.7 102.8 101.1 114.2 143.1 134.395.5 105.5 125.1 97.2 115.3 152.5 125.4 96.9 105.2 117.5130.4 132.8 100.5 120.3 150.4 133.4 163.3 130.6 109.4 130.7157.3 170.3 127.3 118.5 180.3 147.3 127.3 126.8 137.7 106.5143.6 133.5 103.6 166.3 127.4 132.6 140.6 153.6 148.6 143.3137.8 135.4 130.5 177.5 130.8 135.8 127.8 137.8 127.5 93.887.9 96.9 107.9 90.4 119.5 88.9 107.9 127.3 100.9 105.9140.5 120.5 120.8 142.3 123.2 120.8 134.3 82.5 138.3 124.5120.0 125.0 123.1 122.3 113.0 132.4 122.0 129.0 142.6 121.096.8 110.8 106.8 136.5 170.8 117.7 99.8 116.5 136.8 96.494.7 124.1 190.5 165.2 122.4 135.4 137.3 96.3 153.6 133.6107.1 147.8 94.9 127.9 145.5 137.5 142.3 109.0 116.7 125.8

23

imisaTvis, rom movaxdinoT monacemebis dajgufeba, pirvel rigSivnaxoT, Tu rogoria SerCevis diapazoni anu sxvaoba udides da umcir-es monacems Soris: d = x(n)-x(1). Cvens SemTxvevaSi, x(n) = 190.5 da x(1) =

82.5. amitom d = 190.5 - 82.5 = 108.Semdeg, davyoT cvalebadobis intervali, anu [82.5; 190.5] inte-

rvali, k cal qveintervalad [ai-1; ai), i = 1,2,…,k, sadac a0 = x(1) da in-

tervalebis ricxvi k unda akmayofilebs pirobas: ln n k n1/2. Cvens

SemTxvevaSi, n = 100, da maSasadame, intervalebis ricxvi unda akmayof-

ilebdes pirobas 5 k 10. rac Seexeba dayofis klasebs, isini SeiZl-eba iyos Tanabaric da araTanabaric, es damokidebulia Sesaswavli mov-

lenis bunebaze. magaliTad, demografiaSi sikvdilianobis cxrilebis

Sedgenisas, asakobriv intervals [0;5] weli kidev ufro “awvrileben”

erTwlian intervalebad, xolo danarCen asakobriv intervalebs, [5,10),

[10,15), . . . ,[95,100), [100,), ucvlelad toveben. es ganpirobebuliadanarCen asakebTan SedarebiT, mcirewlovanTa maRali sikvdilianobiT

nebismier populaciaSi. magram gacilebiT xSirad iReben Tanabari sigr-

Zis klasebs.

davubrundeT Cvens magaliTs da aviRoT, magaliTad, k = 9. Tanab-ari dayofis SemTxvevaSi, TiToeuli qveintervalis sigrZe tolia l =d/k =108/9=12. maSasadame, pirveli qveintervali iqneba [82.5; 94.5), me-

ore qveintervali [94.5; 106.5) da a.S., bolo qveintervali iqneba

[178.5; 190.5] (SevniSnoT, rom danarCenebisagan gansxvavebiT, bolo int-

ervali Caketilia).

daviTvaloT TiToeul qveintervalSi moxvedril monacemTa rao-denobebi, SevafardoT isini monacemTa saerTo raodenobasTan da Sevad-

ginoT fardobiT sixSireTa Semdegi cxrili:

intervalis nomeri,i

intervali [ai -1;ai )

monacemTa fardobiTi

sixSire, fi

1 [82.5; 94.5) 6/120 = 0.050

2 [94.5; 106.5) 20/120 = 0.167

3 [106.5; 118.5) 17/120 = 0.142

4 [118.5; 130.5) 30/120 = 0.250

5 [130.5; 142.5) 25/120 = 0.210

6 [142.5; 154.5) 13/120 = 0.107

7 [154.5; 166.5) 4/120 = 0.033

8 [166.5; 178.5) 3/120 = 0.025

9 [178.5; 190.5] 2/120 = 0.016

= 1histograma da poligoni monacemebis grafikuli warmodgenis ki-

dev erTi kargi saSualebaa. isini Semdegnairad igeba: TiToeul inter-

valze, rogorc fuZeze, avagoT marTkuTxedi, romlis simaRlea hi = fi /

24

li, li = ai - ai-1. radgan Cvens SemTxvevaSi, li = ai - ai-1 = l =12, amitom hi = fi /12. miRebul marTkuTxedTa erTobliobas histogramas, xolo marTkuT-

xedebis zeda fuZeebis Suawertilebis SemaerTebel texils poligonsuwodeben. Sesabamis grafikebs aqvs Semdegi saxe:

2.2.4. foTlebiani Reroebis msgavsi diagrama

sixSireTa ganawileba grafikulad SeiZleba gamoisaxos agreTvee. w. foTlebiani Reroebis msgavsi diagramiT, rac gulisxmobs mocemu-

li ricxvebidan aTobiTi niSnis gamoyofas (foTlebi) da danarCeni ni-

Snebis Sesabamisi ricxvebis zrdis mixedviT Camoweras TiTojer zevid-

an qvemoT vertikalurad (vertikaluri Rero). Semdeg, vertikalur

Reroze datanili fiqsirebuli pirveli ramodenime saerTo aTobiTi

niSnis gverdiT, am ricxvebis bolo aTobiTi niSnebis (foTlebis) amo-

weras rigrigobiT horizontalurad (horizontaluri Rero).

foTlebiani Reroebis msgavsi diagrama aris monacemTa organize-

bis meTodi, romelic Sedgeba monacemTa daxarisxebis da grafikulad

gamosaxvis kombinaciisagan. dagrovil sixSireTa ganawilebasTan Sedar-ebiT misi upiratesoba imaSia, rom igi ukeT inaxavs faqtiur monacem-

ebs grafikul warmodgenasTan SedarebiT.

foTlebiani Reroebis msgavsi dagrama warmoadgens monacemebis

ganlagebis sqemas (gegmas), romelSic dakvirvebuli TiToeuli monace-

mis erTi nawili gamoiyeneba rogorc vertikaluri Rero, xolo meore

nawili ki rogorc foToli, raTa moxdes jgufebisa da klasebis for-

mireba.

magaliTi 24. qvemoT moyvanilia im pacientebis raodenoba, rom-

lebmac ambulatoriuli mkurnalobis centrs mimarTes kardiogramis

25

gadasaRebad 20 dRiani SerCevis ganmavlobaSi. avagoT am monacemebisaT-vis foTlebiani Reroebis msgavsi diagrama.

25 31 20 32 13

14 43 02 57 23

36 32 33 32 44

32 52 44 51 45

amoxsna.

nabiji 1. davalagoT monacemebi zrdadobis mixedviT:

02, 13, 14, 20, 23, 25, 31, 32, 32, 32,

32, 33, 36, 43, 44, 44, 45, 51, 52, 57.

SeniSvna: monacemebis zrdis mixedviT dalageba ar aris aucilebeli, magram is moxerxebulia foTlebiani Reroe-

bis msgavsi diagramis asagebad.

nabiji 2. ganvacalkevoT (davajgufoT) monacemebi pirveli ci-

fris mixedviT, ise rogorc qvemoTaa naCvenebi:

02 13, 14 20, 23, 25 31, 32, 32,

32, 32, 33, 36

43, 44, 44, 45 51, 52, 57.

nabiji 3. avagoT diagrama, risTvisac monacemis pirveli cif-

ri gamoviyenoT vertikalur Rerod, xolo momdevno

cifri ki – foTlad. magaliTad, monacemisaTvis 32, pirveli cifri 3, aris Rero, xolo meore cifri 2,

aris foToli. monacemisaTvis 14, 1 aris Rero, xolo

4 ki foToli da a. S. sabolood, miviRebT qvemoT

moyvanil diagramas:

0 2

1 3 4

2 0 3 5

3 1 2 2 2 2 3 6

4 3 4 4 5

5 1 2 7agebuli foTlebiani Reroebis msgavsi diagrama gviCvenebs, rom

monacemebis am ganawilebas piki aqvs centrSi da monacemebSi ar aris

wyveta. 20 dRidan 7 dRis ganmavlobaSi im pacientebis raodenoba, ro-

mlebmac gadaiRes kardiograma moTavsebulia 31-dan 36-mde. diagrama ag-

reTve gviCvenebs, rom ambulatoriul centrSi pacientebis raodenoba

dRis ganmavlobaSi meryeobda minimum 2 pacientidan maqsimum 57 pacien-

tamde.

amocanebi1. 25 axalwveuls gaukeTes sisxlis analizi, raTa gaerkviaT

maTi sisxlis jgufi (tipi). monacemebis simravle aseTia:

26

A B B AB OO O B AB BB B O AB OA O O O ABAB A O B A

avagoT sixSireTa ganawileba da wriuli diagrama am monacemebisTvis.

2. qvemoT moyvanili monacemebi warmoadgens rekordul tempera-

turebs aSS-s 50 StatisTvis. avagoT am monacemebisaTvis dajgufebuli

sixSireTa ganawileba 7 klasis gamoyenebiT:

112 100 127 120 134 118 105 110 109 112110 118 117 116 118 122 114 114 105 109107 112 114 115 118 117 118 122 106 110116 108 110 121 113 120 119 111 104 111120 113 120 117 105 110 118 112 114 114

3. qvemoT moyvanili monacemebi warmoadgens milebis raodenobas,

rasac gadis erT galoni benziniT 30 sxvadasxva tipis manqana qalaqSi

moZraobisas. avagoT sixSireTa ganawileba.

12 17 12 14 16 1816 18 12 16 17 1515 16 12 15 16 1612 14 15 12 15 1519 13 16 18 16 14

4. avagoT histograma im monacemebis warmosadgenad, romlebic

moyvanilia qvemoT da Seesabameba amerikis 50-ive Statis rekordulad

maRal temperaturebs:

klasis sazRvrebi sixSire99.5-104.5 2

104.5-109.5 8109.5-114.5 18114.5-119.5 13119.5-124.5 7124.5-129.5 1129.5-134.5 1

5. wina magaliTSi moyvanili sixSireTa ganawilebisaTvis avagoT

sixSireTa poligoni.

6. avagoT histograma da sixSireTa 20 SemTxveviT SerCeuli

morbenalis mier kviris ganmavlobaSi garbenili manZilebis ganawile-

bis mixedviT:

klasis sazRvrebi sixSire dagrovili sixSire5.5--10.5 1 1

10.5--15.5 2 3

15.5--20.5 3 6

20.5--25.5 5 11

27

25.5--30.5 4 15

30.5--35.5 3 18

35.5--40.5 2 20

7. qvemoT moyvanilia 1998 wlis ganmavlobaSi msubuq sauzmeze

daxarjuli funtebis raodenobis ganawileba sakvebis saxeobebis mixed-

viT. avagoT marTkuTxedebiani da wriuli diagrama.

msubuqi sauzme funtebi (sixSire)

kartofilis “Cifsi” 11.2 milioni

simindis “Cifsi” 8.2 milioni

funTuSa 4.3 milioni

simindis burbuSela 3.8 milioni

keqsi 2.5 milioni

jami n 30.0 milioni

8. sadazRvevo kompaniis xelmZRvanelobas ainteresebs gamoikvli-os gasuli zafxulis 30 dRis ganmavlobaSi did qalaqSi moparul av-

tomanqanaTa ganawileba. dakvirvebuli nedli monacemebi moyvanilia qve-

moT:52 62 51 50 69

58 77 66 53 5775 56 55 67 73

79 59 68 65 72

57 51 63 69 7565 53 78 66 55

avagoT foTlebiani Reroebis msgavsi diagrama Semdegi klasebis gamoye-

nebiT: 50—54, 55—59, 60—64, 65—69, 70—74, 75—79 da gavakeT-

oT Sesabamisi daskvnebi.

l e q c i a 2.

Tavi 3. ricxviT monacemTa SerCeviTi maxasiaTeblebi.

3.1. saSualo yofaqcevis maxasiaTeblebi.

3.1.1. SerCeviTi saSualo (saSualo ariTmetikuli).

SerCeviTi saSualo warmoadgens SerCevaSi Semavali monacemebis

saSualo ariTmetikuls. is aRiniSneba nx simboloTi da maTematikur-

ad ase Caiwereba:

nx = (1/n) (x1 + x2 +…+ xi )

n

iix

n 1

1. (3.1)

A axalSobilTa wonebis 2.1 magaliTisaTvis gvaqvs:

20x =

20

120

1

iix = (0.05) (x1 + x2 +…+ x20 ) =

28

(0.05)(3265+3260+3245+3484+4146+3323+3649+3200+3031+2069+2581++2841+3609+2838+3541+2759+3248+3314+3101+2834) = 3166.9 (grami).

SevniSnoT, rom saSualo ariTmetikulis gamoTvlisaTvis monac-

emebis Tanmimdevrobas araviTari mniSvneloba ara aqvs. ase, rom

n

iin x

nx

1)(

1. (3.2)

monacemTa toli mniSvnelobebis SemTxvevaSi, ufro moxerxebu-lia SerCeviTi saSualos gamoTvli Semdegi wesi: variantebi (gansxva-vebul monacemTa mniSvnelobebi) gavamravloT Sesabamis sixSireebze(maT raodenobebze SerCevaSi), SevkriboT da miRebuli Sedegi gavyoT

monacemTa raodenobaze, anu

r

ii

in nx

nx

1

)(1, (3.3)

sadac x(i) aRniSnavs i-uri variantis ricxviT mniSvnelobas, ni – i-urivariantis sixSires da r - variantebis raodenobas.

SevniSnoT, rom SerCeviTi saSualo warmoadgens jams varianteb-is mniSvnelobebis namravlisa Sesabamis fardobiT sixSireebze

r

ii

in fxx

1

)( , (3.4)

sadac fi aRniSnavs i-uri variantis fardobiT sixSires fi = ni /n. amave wesiT gamoiTvleba SerCeviTi saSualo dajgufebuli mona-

cemebis SemTxvevaSic, oond x(i) variantis ricxviT mniSvnelobis magiv-

rad, (3.4) formulaSi iReben dajgufebis i-uri intervalis Sua wert-

ils kiaam iii ,...,2,1,2/)( 1 :

r

iiin fmx

1

. (3.5)

(3.5) formulis gamoyenebis sailustraciod gavixsenoT fardo-

biT sixSireTa cxrili 2.3 magaliTidan. am SemTxvevaSi, k = 9. cxrili-

dan gvaqvs, rom 5.882/)5.945.82(2/)( 101 aam . analogiurad,

m2 = 100.5, m3 = 112.5, m4 = 124.5, m5 = 136.5, m6 = 148.5, m7 = 160.5, m8 =172.5, m9 = 184.5. amitom (3.5) formulidan miviRebT:

9x 88.50.05+100.50.167+112.50.142+124.50.25+136.50.21+148.50.107++160.50.033+172.50.025+184.50.016 = 125.42.

3.1.2. SerCeviTi mediana.

xSirad monacemebis saSualo mniSvnelobas axasiaTeben ara misi

sididis, aramed saSualedo (medianuri) adgilmdebareobis mixedviTac

variaciul mwkrivSi. Sesabamis sidides SerCeviTi mediana ewodeba. Ser-CeviTi medianis gamoTvlis wesi Semdegia:

29

luwiaTu

kentiaTu

nxx

nxm nn

n

n ,2

,

)12()2(

)2)1((

magaliTi 3.1. cxrilSi mocemulia sisxlSi TeTri nawilakebis

raodenobebi cxra avadmyofisaTvis:

I 1 2 3 4 5 6 7 8 9

xI 7000 35000 5000 9000 8000 3000 10000 12000 8000

SevadginoT variaciuli mwkrivi:3000, 5000, 7000, 8000, 8000, 9000, 10000, 12000, 35000.

movZebnoT am cxra mniSvnelobidan Sualeduri, anu iseTi, romlis mar-

cxniv da marjvniv moxvdeba variaciuli mwkrivis wevrTa Tanabari rao-

denoba. radgan wevrTa raodenoba kenti ricxvia, cxadia, rom aseTi iq-

neba variaciuli mwkrivis mexuTe elementi, anu x(5) = 8000. SerCeviT

medianas mn simboloTi aRniSnaven. maSasadame, Cveni magaliTisaTvis, m9

= 8000. gavixsenoT 2.1 magaliTi, sadac gvqonda monacemTa luwi raoden-

oba (n = 20). rogorc vnaxeT, Sesabamis variaciul mwkrivs hqonda Sem-

degi saxe:2069, 2581, 2759, 2834, 2838, 2841, 3031, 3101, 3200, 3245,

3248, 3260, 3265, 3314, 3323, 3484, 3541, 3609, 3649, 4146.

medianis gamosaTvleli formulidan n = 20-Tvis miviRebT:m20 = (x(10)+ x(11))/2 = (3245 + 3248)/2 = 3246.5.

3.1.3. wanacvlebuli (asimetriuli) da simetruli SerCevebi.

ganvixiloT SerCeviTi saSualosa da medianis Sedarebis sakiTxi.

davaxasiaToT nn mx sxvaoba. ganvixiloT jer kenti n-is SemTxveva.

cxadia, rom

n

inin

n

iinn xx

nxx

nmx

1)2/)1(()()2/)1((

1)(

11

2/)1(

1)()2/)1((

12/)1()2/)1(()(

1 n

iin

n

nini xxxx

n.

amrigad, nn mx 0

2/)1(

1)()2/)1((

12/)1()2/)1(()(

n

iin

n

nini xxxx .

maSasdame, nn mx sxvaoba dadebiTia, anu nn mx , Tu medianis marjvn-

iv moTavsebuli monacemebi (mTlianobaSi) ufro metadaa daSorebuli

medianidan, vidre mis marcxniv moTavsebuli monacemebi da piriqiT,

nn mx sxvaoba uaryofiTia, anu nn mx , Tu medianis marcxniv moTav-

sebuli monacemebi ufro metadaa daSorebuli medianidan, vidre mis ma-

rjvniv moTavsebuli monacemebi. analogiuri msjeloba samarTliania

30

luwi n-isaTvisac. pirvel SemTxvevaSi amboben, rom SerCeva dadebiTad,xolo meoreSi – uaryofiTadaa wanacvlebuli (asimetriulia). im SemT-xvevaSi, roca SerCeviTi saSualo da mediana daaxloebiT tol mniSvne-

lobebs Rebulobs, vityviT, rom SerCeva simetrulia.

3.1.4. SerCeviTi moda.

SerCeviT monacemebs Soris yvelaze xSirs SerCeviTi moda ewod-eba. 2.1 magaliTis mixedviT, yvelaze xSiri orbavSviani ojaxebia, maTi

ricxvi 23-s Seadgens. aseT SemTxvevaSi (roca arsebobs erTi yvelaze

xSiri monacemi) vityviT, rom SerCeva unimodaluria. Tu SerCevaSiyvelaze xSiri monacemebis raodenoba oria, amboben, rom SerCeva bimod-aluria. Tu xSiri monacemebis raodenoba samia, amboben, rom SerCevatrimodaluria da a.S.

3.2. SerCevis gafantulobis (ganfenilobis) maxasiaTeblebi.

3.2.1. SerCevis diapazoni. boqsploti. rangi.

SerCevis diapazoni ewodeba sxvaobas SerCevis udides da umcir-

es mniSvnelobebs Soris: d =xmax-xmin=x(n)-x(1).

2.3 magaliTis SemTxvevaSi, x(n) = 190.5 da x(1) = 82.5. amitom SerC-evis diapazoni tolia d = 190.5 - 82.5 = 108.

SerCeviTi diapazonis ricxviTi mniSvneloba, roca is sakmarisaddidia, bevrs verafers gveubneba dakvirvebadi movlenis Sesaxeb. garda

amisa, is arsebiTadaa damokidebuli SerCevis n moculobaze da ar gam-

odgeba sxvadasxva moculobis SerCevaTa Sesadareblad.

saSualedo monacemebis ganlagebisa da Sesabamisad, maTi gafant-

ulobis Sesaxeb informacias iZleva e. w. procentilebi (percentiles).procentilebi erTdroulad warmoadgens monacemTa ganlagebisa da ma-

Ti gafantulobis sazoms. vTqvaT, P raime ricxvia, moTavseebuli 0-sa

da 100-s Soris 0 100P . monacemTa simravlis P rigis procenti-

li (ubralod P -procentili) ewodeba iseT px mniSvnelobas (sidid-

es), romelsac gaaCnia Semdegi Tviseba: monacemTa araumetes P %-isa

naklebia an toli px -ze da araumetes (100-P )%-isa metia an toli

px -ze. cxadia, rom mediana warmoadgens 50-procentils. Tu 100P ,

maSin medianis analogiurad P -procentilis ( px -s) gamosaTvleli gam-

osaxuleba iqneba:

([ ] 1)p nx x , roca n ar aris mTeli ricxvi, da

( ) ( 1)

2n n

p

x xx , roca n mTeli ricxvia.

25-procentils ewodeba pirveli kvartili (quartile) da aRiniS-

neba 1Q -iT (10-procentils decili (decile) ewodeba), xolo 75-proce-

31

ntils ewodeba mesame kvartili da aRiniSneba 3Q -iT. 50-procentils

(romelic warmoadgens medianas), agreTve uwodeben meore kvartils.

sxva sityvebiT, rom vTqvaT, 1 2( , , )Q x Q kvartilebi monacemTa simravles

pirobiTad oTx tol nawilad yofs. kvartilTSorisi gabnevis diapaz-

oni (Interquartial range -- IQR) ewodeba sidides: 3 1IQR Q Q . kvarti-

lebi medianasTan erTad, e. i. sameuli 1 2( , , )Q x Q gvaZlevs garkveul

warmodgenas monacemTa centris, gabnevisa da ganawilebis (gagluvebuli

histogramisa da poligonis) formis Sesaxeb. am SemTxvevaSi

3 113 6Q x x Q (mesame kvartili ufro Sorsaa medianisagan, vid-

re pirveli kvartili), rac miuTiTebs imaze, rom ganawileba marjvnivasimetriulia (misi marjvena bolo ufro asimetriulia, vidre marcxe-

na).

ukanasknel periodSi gansakuTrebuli popularobiT sargeblobs

monacemTa ganawilebis vizualuri warmoCenis axali saSualeba – e. w.

boqsploti (boxplot). boqsploti mdgradi sazomebis – ( 1Q , x , 3Q ,

IQR ) meSveobiT saSualebas iZleva gamoikveTos monacemTa simravlis

zogierTi Tavisebureba: a). ganlagebis centri; b). gafantuloba; g). ga-

vrcoba da simetriulobidan gadaxris buneba da bolos, moxdes d).

“amovardnili (outlier)” monacemebis identifikacia. aRvweroT, Tu rog-

or xdeba boqsplotis ageba:3IQR 3IQR

1.5IQR 1.5IQR

Q1x Q3

boqsploti

1) gavavloT horizontaluri RerZi da masze movniSnoT 1Q , x da 3Q ;

2) avagoT marTkuTxedi, romlis gverdebi marTobulia horizontaluri

RerZis da romlis marcxena gverdi 1Q -is, xolo marjvena ki 3Q -is

zemoTaa moTavsebuli;

3) gavavloT vertikaluri wrfe marTkuTxedSi medianis zeviT;

4) marTkuTxedis TiToeuli gverdidan gavavloT horizontaluri Rer-

Zis paraleluri ori wrfe, pirvelze gadavzomoT 1.5 IQR , xolo meor-

eze ki 3 IQR sigrZis monakveTebi;

5) yoveli monacemis Sesabamis wertilze, romelic marTkuTxedis gver-

didan gadazomil 1.5 IQR -sa da 3 IQR -s Sorisaa moTavsebuli Semovxaz-

oT wrewiri. am monacemebs ewodeba zomieri “amovardnebi”, xolo im

monacemebis Sesabamis wertilebze, romlebic moTavsebulia marTkuTxe-

32

dis gverdidan gadazomili 3 IQR -is gareT SemovxazoT gamuqebuli wre.

aseTi wreebiT Semoxazul monacemebs eqstremaluri “amovardnebi” ewo-

deba.

monacemis rangi (rank) ewodeba am monacemis nomers variaciul

mwkrivSi (toli monacemebis SemTxvevaSi TiToeuls mivaniWoT maTze

mosuli rangebis saSualo ariTmetikuli). mocemuli monacemis procen-tuli rangi warmoadgens am monacemis qveviT mdgom monacemebze mosu-

li procentebisa (1

100%r

n

) da Tavad monacemze mosuli procenteb-

is naxevaris (0.5

100%n ) jams. Tu n SerCevis moculobaa, xolo mona-

cemis rangia r , maSin am monacemis procentuli rangi moicema Semdegi

formuliT:1 0.5 2 1

100% 100% 100%2

r r

n n n

. Tu SerCevis elementis

procentuli rangia P , maSin es elementi warmoadgens SerCevis P -

procentils

3.2.2. SerCeviTi absoluturi gadaxra.

absoluturi gadaxris azriT monacemebis mniSvnelobaTa gafant-

ulobis saukeTeso maCvenebels warmoadgens sidide

n

ini mx

n 1

||1

, ro-

melsac SerCeviT absolutur gadaxras uwodeben. es sidide gacilebiT

“mdgradia” anomaluri monacemebis SemTxvevaSi, vidre saSualo standa-

rtuli gadaxra (ix. qvemoT), Tumca garkveuli mosazrebebis gamo, mas

praqtikaSi naklebad iyeneben.

3.2.3. SerCeviTi dispersia. SerCeviTi standartuli gadaxra.

SerCeviTi dispersia warmoadgens monacemTa gafantulobis saz-

oms, romelic gviCvenebs monacemebis mniSvnelobaTa saSualo gadaxrasSerCeviTi saSualosagan:

n

inin xx

ns

1

22 )(1

. (3.6)

ariTmetikul kvadratul fesvs SerCeviTi dispersiidan SerCeviT

standartul gadaxras uwodeben da mas n -iT aRniSnaven:

n =

n

ini xx

n 1

2)(1

. (3.7)

axalSobilTa wonebis Sesaxeb 2.1 magaliTisaTvis gvaqvs220s = (1/20)((3265 – 3166.9)2 + (3260 – 3166.9)2 + + (2834 – 3166.9)2 )

188452; 20 = (188452)1/2 434.3 (grami).

33

garkveuli mosazrebebis gamo (ix. leqcia 7), 2ns –isa da n -is

nacvlad iyeneben e.w. Sesworebul SerCeviT dispersiasa da Sesworebulstandartul gadaxras. es Sesworeba gulisxmobs zemoxsenebuli sidi-

deebis gamosaxulebebSi kvadratebis jamis gayofas (n -1)–ze da ara n-ze. miRebuli sidideebi Sesabamisad 2

ns –iTa da n-iT aRiniSneba. maSas-

adame,

2ns =

n

ini xx

n 1

2)(1

1da n =

n

ini xx

n 1

2)(1

1. (3.8)

2.1 magaliTisaTvis gveqneba, rom:220s = (1/19)((3265 – 3166.9)2 + (3260 – 3166.9)2 + … + (2834 – 3166.9)2 )

207979.2; 20 = (207979.2)1/2 456.05 (grami).

am maxasiaTebelTa mniSvnelovnebaze laparakobs Semdegi magaliTi.

magaliTi 3.2. qvemoT moyvanilia erTidaimave pirovnebisaTvis qo-

lesterinis sididis ori sxvadasxva meTodiT gazomvis Sedegebi:

avtoanalizuri meTodi

(mg%/ml)

mikroencimaluri meTodi

(mg%/ml)

cxadia, rom

5x = (1/5) (177 + 193 + 195 + 209 + 226) = 200,

5y = (1/5) (192 + 197 + 200 + 202 + 209) = 200,

anu orive meTodiT miRebuli SerCeviTi saSualoebi tolia, maSin ro-

desac vizualuradac kargad Cans, rom es SerCevebi radikalurad gans-xvavdebian: meore SemTxvevaSi monacemebi ufro mWidrodaa Tavmoyrili

SerCeviTi saSualos mimarT, vidre pirvelSi. es aisaxeba swored am

monacemebis SerCeviT dispersiebSi (standartul gadaxrebSi):

avtoanalizuri meTodis SemTxvevaSi gvaqvs

5=((1/4)((177-200)2+(193-200)2+(195-200)2+(209-200)2+(226-200)2))1/2=3401/2 18.4,

xolo mikroencimaluri meTodis SemTxvevaSi gveqneba

5=((1/4)((192-200)2+(197-200)2+(200-200)2+(202-200)2+(209-200)2))1/2=(39.5)1/26.3.

rogorc vxedavT, meore SemTxvevaSi standartuli gadaxra (gaf-

antuloba) pirvelTan SedarebiT samjer naklebia da bunebrivia CavTva-

loT, rom adamianis organizmSi qolesterinis Semcvelobis dasadgenad

200

177 193 195 209 226

192 197 202 209

34

meore meTodi jobia pirvels, radgan am SemTxvevaSi naklebia gazomvis“Secdomis riski”.

3.2.4. variaciis koeficienti.

SerCeviTi variaciis koeficienti, Vn simboloTi aRiniSneba da

ganimarteba rogorc SerCeviTi standartuli gadaxris Sefardeba Ser-

CeviTi saSualos ricxviT mniSvnelobasTan, anu

n

nn x

V

. (3.9)

gamovTvaloT variaciis koeficientis ricxviTi mniSvneloba axa-

lSobilTa wonebis 2.1 magaliTSi. rogorc gvaxsovs, 20x =3166.9 da

20 =456.05. amitom V20 = 456.05 / 3166.9 = 0.144 (an procentebSi 14.4%).

variaciis koeficienti gansakuTrebiT xSirad gamoiyeneba im

sxvadasxva SerCevebis variaciulobis (cvalebadobis) Sesadareblad,

romlebsac gansxvavebuli SerCeviTi saSualoebi aqvT.

sailustraciod ganvixiloT Semdegi magaliTi. cnobilia, rom

axalSobilebi pirveli ramdenime dRis ganmavlobaSi ikleben wonaSi da

mxolod amis Semdeg xdeba maTi wonaSi mateba. 2.1 magaliTSi moyvanil

monacemebTan erTad, SevxedoT axalSobilTa wonebs dabadebidan sami

dRis Semdeg da davinteresdeT Tu ramdenad cvalebadobs monacemebi.

cxrilSi mocemulia axalSobilTa wonebi (2.1 magaliTis mixedviT) da

maTi wonebi sami dRis Semdeg (ujredebis qveda striqonSi mocemuliricxvebi):

i xi i xi i xi i xi

132653127

633233205

1125812345

1627592700

232603053

736493407

1228412710

1732483155

332453180

832003054

1336093500

1833143265

434843350

930312827

1428382709

1931013010

541464010

1020691970

1535413480

2028342782

rogorc gvaxsovs, 20x = 3166.9, 20 = 456.05 da V20 = 0.144.

msgavsi gamoTvlebi ujredebis qveda striqonSi mdgomi monacemebisaT-

vis iZleva: 120x =3042, 1

20 = 445.46 da 120V = 0.146. rogorc vxedavT,

miuxedavad imisa, rom axalSobilTa wonam sami dRis ganmavlobaSi sa-

Sualod daiklo 125 (=3166.9-3042) gramiT, variaciuloba TiTqmis ar

Secvlila V20 120V .

35

variaciis koeficientis gamoyeneba mizanSewonilia agreTve sxva-dasxva sidideebze dakvirvebebiT miRebuli SerCevebis Sesadareblad.

3.3. SerCeviTi ricxviTi maxasiaTeblebis Tvisebebi.

3.3.1. SerCeviTi saSualos Tvisebebi:

1) davuSvaT SerCevis yoveli monacemi wavanacvleT erTidaigive

c1 sididiT, anu yi = xi + c1, i = 1,2,…,n, maSin miRebuli SerCevis SerCev-

iTi saSualo wanacvldeba c1-is toli sididiT, anu 1cxy nn . marT-

lac,

)...(1

)(11

112111

11

cxcxcxn

cxn

yn

y n

n

ii

n

iin

1121121

1)...(

1)...(

1cxcn

nxxx

ncnxxx

n nnn

2) davuSvaT SerCevis yovel monacems SevucvaleT masStabi erT-

idaigive c2 sididiT, anu yi = c2xi, i = 1,2,…,n. maSin miRebuli SerCevis

SerCeviTi saSualos masStabic Seicvleba c2-jer. marTlac,

)...(111

222121

21

n

n

ii

n

iin xcxcxc

nxc

ny

ny

nn xcxxxn

c 221

2 ... .

SevniSnoT, rom Tu xi = c, i = 1,2,…,n, maSin cxn , anu mudmivi

SerCevis SerCeviTi saSualo TviTon es mudmivia.

gavaerTianoT wina ori Tviseba Semdeg erT Tvisebad:

3) Tu gardavqmniT SerCevis yovel monacems erTidaigive wrfivi

gardaqmniT yi = c2xi + c1, i = 1,2,…,n, maSin SerCeviTi saSualoc Seicvle-

ba imave wrfivi gardaqmniT, anu 12 cxcy nn .

SesaZlebelia ukanasknelis kidev ufro ganzogadeba. kerZod, sam-

arTliania Semdegi Tviseba:

4) davuSvaT, SerCeva warmoadgens ori (an meti) sxvadasxva Ser-

Cevis wrfiv kombinacias, anu zi = c1xi + c2 yi, yoveli i-saTvis, i = 1,2,…,n, maSin SerCeviTi saSualoc iqneba saSualoebis Sesabamisi wrfivi ko-

mbinacia: nnn ycxcz 21 .

5) ori SerCevis gaerTianebiT miRebuli SerCevis saSualo cal-keul saSualoTa Sewonili jamis tolia.

Tu 1x , . . . , nx da 1y , . . . , my ori SerCevaa, maSin 1x , . . . ,

nx , 1y , . . . , my gaerTianebuli SerCevis saSualo moicema formuliT:

n mz x y

n m n m

.

36

3.3.2. SerCeviT dispersiis Tvisebebi:1) mudmivi SerCevis SerCeviTi dispersia (da standartuli gad-

axra) 0-is tolia. marTlac, radganac xi = c, i = 1,2,…,n, da cxn , ami-

tom

2ns = 00

1

1)(

1

1)(

1

1

11

2

1

2

n

i

n

i

n

ini n

ccn

xxn

.

2) davuSvaT SerCevis yoveli monacemi wavanacvleT erTidaigive

c1 sididiT, anu yi = xi + c1, i = 1,2,…,n, maSin miRebuli SerCevis SerCev-

iTi dispersia ar Seicvleba, anu 2,nys = 2

,nxs .

3) davuSvaT SerCevis yovel monacems SevucvaleT masStabi erT-

idaigive c sididiT, anu yi = cxi, i = 1,2,…,n. maSin miRebuli SerCevis

SerCeviTi dispersia Seicvleba c2 sididiT, anu 2,nys = c2 2

,nxs . marTlac,

2,nys

n

ini yy

n 1

2)(1

1

2

1

22

1

2 )(1

1)(

1

1cxx

ncxcxc

n

n

ini

n

ini

2,nxs .

4) Tu 1,..., nx x da 1,..., my y ori SerCevaa, maSin gaerTianebuli

SerCevisaTvis 1 1,..., , ,...,n mx x y y SerCeviTi dispersia da Sesworebuli

SerCeviTi dispersiebi gadaiTvleba Semdegnairad:

2 2 2 22

( )( )x y

n m mns s s x y

m n m n m n

da

'2 '2 '2 21 1( )

( 1)( )x y

n m mns s s x y

m n m n m n m n

(im kerZo SemTxvevaSi, roca m n da x y , gaerTianebuli SerCevis

SerCeviTi dispersia iqneba Tavidan aRebuli SerCevebis SerCeviTi dis-

persiebis saSualo ariTmetikuli: 2 2 2( ) / 2x ys s s ).

3.4. dawyvilebuli monacemebi. korelacia

xSir SemTxvevaSi mkvlevari dainteresebulia daadginos or si-

dides Soris kavSiri arsebobs Tu ara. am mizniT ori damkvirvebeli

erTdroulad atarebs dakvirvebebs da miRebul monacemebs awyvilebs

erTmaneTTan. magaliTad, Tu Cven gvainteresebs kavSiri haeris tempera-turasa da atmosferul wnevas Soris, maSin drois erTi da imave mom-

entSi pirveli damkvirvebeli zomavs haeris temperaturas, xolo meo-

re ki atmosferul wnevas. pirveli dakvirvebis an gazomvis obieqts

uwodeben damoukidebel sidides, da aRniSnaven x -iT, xolo meore dak-

virvebis an gazomvis obieqts uwodeben damokidebul sidides da aRni-

37

Snaven y -iT. ama Tu im or maxasiaTebels (an or maCvenebels) Soris

kavSiris Seswavlis mizniT am maxasiaTeblebis dakvirvebuli mniSvnel-

obebis simravle warmovadginoT wyvilebis simravlis saxiT: 1 1( , ),x y

2 2( , )x y , . . . , ( , )n nx y . am monacemebis mixedviT ageben e. w. gabnevis dia-

gramas. x da y sidideebs Soris SesaZlebelia arsebobdes ramodenime

gansxvavebuli tipis damokidebuleba. am damokidebulebis garkveva (id-

entificireba) SesaZlebelia sakoordinato sibrtyeze wertilebis gan-

lagebis struqturaze dayrdnobiT:

1. dadebiTi wrfivi kavSiri arsebobs im SemTxvevaSi, rodesac

wertilebi daaxloebiT lagdeba aRmavali swori xazis irgvliv.

2. uaryofiTi wrfivi kavSiri arsebobs im SemTxvevaSi, rodesac

wertilebi daaxloebiT lagdeba daRmavali swori xazis irgvliv.

3. arawrfivi kavSiri arsebobs im SemTxvevaSi, rodesac werti-

lebi daaxloebiT lagdeba arawrfivi wiris irgvliv.

4. ar aris kavSiri im SemTxvevaSi, rodesac wertilebi uwesri-godaa mimofantuli, ar Cans rom isini raime wiris irgvlivaa koncen-

trirebuli.

dadebiTi wrfivi kavSiri

0

100200

300

400

0 50 100 150

uaryofiTi wrfivi kavSiri

0

100

200

300

400

0 50 100 150

arawrfivi kavSiri

0

100

200

300

400

0 50 100 150

ar aris kavSiri

0

200

400

600

0 50 100 150

wirs, romelic aRwers kavSirs (damokidebulebas) x da y si-dideebis mniSvnelobebs Soris misadagebis wiri ewodeba. am wiris pov-

nis sakiTxebs swavlobs maTematikuri statistikis nawili, romelsac

regresiuli analizi ewodeba. wiris mosaZebnad iyeneben e. w. umcires

38

kvadratTa meTods, romlis Tanaxmadac iZebneba iseTi wiri ( )y f x ,

rom: gamosaxuleba 2

1

[ ( )]n

i ii

y f x

aRwevs Tavis minimums.

Tu magaliTad, sidideebs Soris gvaqvs wrfivi kavSiri, maSin

misadagebis wiri wrfivi funqciaa, anu ( )f x ax b ,sadac

1 1 1

2 2

1 1

( ) ( ) ( )

( ) ( )

n n n

i i i ii i i

n n

i ii i

n x y x ya

n x x

da

2

1 1 1 1

2 2

1 1

( ) ( ) ( ) ( )

( ) ( )

n n n n

i i i i ii i i i

n n

i ii i

y x x x yb

n x x

.

magaliTi 3.3. avagoT gabnevis diagrama monacemebisaTvis, romel-

ic miiReba 6 SemTxveviT SerCeuli pirovnebis asaksa da arteriuli

wnevas Soris damokidebulebis Seswavlisas. monacemebis cxrilia:

persona asaki, x wneva, y

A 43 128

B 48 120

C 56 135

D 61 143

E 67 141

F 70 152

amoxsna.

nabiji 1. davxazoT sakoordinato sibrtye da movniSnoT x day RerZebi.

nabiji 2. movniSnoT TiToeuli wertili grafikze rogorc es

qvemoTaa naCvenebi:

gabnevis diagrama

110

120

130

140

150

160

30 40 50 60 70 80

asaki

wneva

statistikaSi iyeneben e. w. korelaciis koeficients, raTa gaz-omon or sidides Soris damokidebulebis xarisxi. SerCeviTi korelac-

iis koeficienti moicema Semdegi TanafardobiT:

1 1 1

2 2 2 2

1 1 1 1

( ) ( ) ( )

[ ( ) ( ) ] [ ( ) ( ) ]

n n n

i i i ii i i

n n n n

i i i ii i i i

n x y x yr

n x x n y y

.

39

magaliTi 3.4. gamovTvaloT korelaciis koeficientis mniSvnel-oba 3.3 magaliTSi.

amoxsna.

nabiji 1. gavakeToT cxrili ise rogors es qvemoTaa naCvenebi:

persona asaki, x wneva, y xy 2x 2y

A 43 128

B 48 120

C 56 135

D 61 143

E 67 141

F 70 152

nabiji 2. gamovTvaloT xy , 2x da 2y gamosaxulebaTa mniSvne-

lobebi da SevitanoT cxrilis Sesabamis svetebSi.

SevkriboT svetebSi moTavsebuli mniSvnelobebi. das-

rulebul cxrils eqneba Semdegi saxe:

pers-ona

asaki, x wneva, y xy 2x 2y

A 43 128 5.504 1.849 16.384

B 48 120 5.760 2.304 14.400

C 56 135 7.560 3.136 18.225

D 61 143 8.723 3.721 20.449

E 67 141 9.447 4.489 19.881

F 70 152 10.640 4.900 23.104

345x 819y xy

47.634

2x

20.399

2 112.443y

nabiji 3. miRebuli mniSvnelobebi SevitanoT korelaciis koe-

ficientis formulaSi da gamovTvaloT r :

2 2

6 47.634 345 8190.897

(6 20.399 345 ) (6 112.443 819 )r

.

daskvna. korelaciis koeficientis miRebuli mniSvneloba gveubn-

eba, rom adamianis asaksa da arteriul wnevas Soris arsebobs mkacrad

dadebiTi wrfivi damokidebuleba.

amocanebi1. cxrilSi mocemulia 1033 axalwveulis astigmatizmis doneTa

sixSiruli ganawileba:

Aastigmatizmis doneebi (diopterebi) sixSireebi0.0 an 0.2-ze naklebi 458

40

0.2 – 0.3 2680.4 – 0.5 1510.6 – 1.0 791.1 – 2.0 442.1 - 3.0 19 3.1 - 4.0 94.1 – 5.0 35.1 – 6.0 2

= 10331.1.AaageT histograma da poligoni;

1.2. gamoTvaleT SerCeviTi (dajgufebuli) saSualo da standartuli

gadaxra.

2. cxrilSi mocemulia avadmyofTa qolesterinis sidideebi die-tamde da dietis Semdeg.

Mmonacemi dietis win dietis Semdeg sxvaoba

1 195 146 492 145 155 -103 205 178 274 159 146 135 244 208 366 166 147 197 250 202 488 236 215 219 192 184 810 224 208 1611 238 206 3212 197 169 2813 169 182 -1314 158 127 3115 151 149 216 197 178 1917 180 161 1918 222 187 3519 168 176 -820 168 145 2321 167 154 1322 161 153 823 178 137 4124 137 125 12

2.1. gamoTvaleT qolesterinis saSualo cvlileba;

2.2. gamoTvaleT qolesterinis cvlilebis saSualo standartuli gad-

axra;

41

2.3. gamoTvaleT cvlilebis SerCeviTi mediana;2.4. simetrulia Tu ara qolesterinis cvlileba?

2.5. zogierTi mkvlevaris azriT, dietis efeqturoba ufro cxadia ma-

Rali qolesterinis mqone pacientebs Soris, vidre dabali qolesteri-

nis Semcvelobis pacientebisaTvis. SegiZliaT Tu ara raimes Tqma amis

Sesaxeb, Tuki monacemebs dayofT medianaze maRal da dabal monacemeb-

ad?

3. imis dasadgenad, moqmedebs Tu ara adamianis mdgomareoba wnev-

aze, Caatares Semdegi eqsperimenti: gauzomes sisxlisa da gulis wne-

va 25 adamians rogorc mwoliare mdgomareobaSi, ise fexze mdgomebs

gaSlili mklavebiT. monacemebi warmodgenilia cxrilSi (orive SemTxv-evaSi pirveli Seesabameba sisxlis, xolo meore gulis wnevas):

sisxlis wneva (mm vwy.sv.)

monacemi mwoliare mdgomareobaSi Fmdgar mdgomareobaSi

1 99 71 105 792 126 74 124 763 108 72 102 684 122 68 114 725 104 64 96 626 108 60 96 567 116 70 106 708 106 74 106 769 118 82 120 9010 92 58 88 6011 110 78 102 8012 138 80 124 7613 120 70 118 8414 142 88 136 9015 118 58 92 5816 134 76 126 6817 118 72 108 6818 126 78 114 7619 108 78 94 7020 136 86 144 8821 110 78 100 6422 120 74 106 7023 108 74 94 7424 132 92 128 8825 102 68 96 64

3.1. gamoTvaleT sxvadasxva mdgomareobaSi gulisa da sisxlis wnevebissxvaobaTa SerCeviTi saSualoebi da standartuli gadaxrebi.

42

4. stefanos kunZulebidan Camotanili Sphenodou punctatus-s kve-rcxebi moTavses sam inkubatorSi, romlebSic Sesabamisad iyo 18, 21 da

22 gradus celsiusis toli mudmivi temperatura. cxrilSi moyvani-

lia gamoCekili wiwilebis wonebi:

inkubatori # wiwilebis wona

I 80 82 81 82 73 85 81 83 80 82

II 91 84 83 90 83 89 85 82 84 87

III 81 83 84 87 76 84 88 83 91 84

4.1. aageT foTlebiani Reroebis msgavsi sami diagrama da SeadareT er-

TmaneTs;4.2. axdens Tu ara inkubatoris temperatura gavlenas wiwilebis won-

ebze?

4.3. gamoTvaleT da SeadareT erTmaneTs SerCeviTi maxasiaTeblebi.

5. Rostkovia magelanica xarobs mxolod centralur otagoSi da

takaCeSi. cxrilSi moyvanilia am mcenareebis foTlebis sigrZeebi mxar-

eebis mixedviT:

12.7 14.4 13.0 12.6 12.8 13.3 12.4 13.5 13.4 14.3otago

14.3 13.9 13.2 13.3 12.7 12.9 13.7 12.9 13.7

12.6 12.7 13.6 13.1 12.3 12.8 11.5 12.3 13.0 12.5takaCe

12.7 13.3 12.5 13.1

5.1. aageT foTlebiani Reroebis msgavsi ori diagrama. gamoiyeneT erTi-

daigive Rero, pirveli simravlis ricxvebis foTlebi moaTavseT Rero-

dan marcxniv, meore simravlis ricxvebis foTlebi ki marjvniv.

5.2. gansxvavdebian Tu ara es simravleebi erTmaneTisagan?

5.3. SeadareT erTmaneTs am monacemebis saSualoebi, medianebi da stan-

dartuli gadaxrebi.

6. qvemoT moyvanilia mdinare viaraus napiris 30 kvadratul me-

trian ubanze Pimelia concinna-s buCqebis raodenobebi:

buCqebis

raodenoba

0-4 5-9 10-14 15-19 20-

24

25-

29

30-35 sul

ubnebis

raodenoba

8 5 6 4 3 3 1 30

6.1. saSualod ramdeni buCqia ubanze?6.2. aageT poligoni;

6.3. SeafaseT am monacemebis cvalebadoba;

6.4. Tu Tqveni mizania es monacemebi SeadaroT sxva mdinaris napiris

52 ubanze gazrdili buCqebis raodenobas, ra ufro ukeTesia aagoT:

sixSireTa poligoni, Tu fardobiT sixSireTa poligoni? axseniT Tqve-

ni pasuxi.

7. botanikosma daTvala yoveli darguli 12 xidan ramdeni gax-

ma:

43

gamxmari xeebisraodenoba

0 1 2 3 4 5 6 7 8 9 10 11 12

sixSire 7 15 20 11 6 0 1 1 1 1 0 0 0

7.1. sul ramdeni xe daaTvaliera botanikosma?

7.2. saSualos 12-dan ramdeni xe gaxma?

8. 20-ma studentma Sualedur gamocdaze miiRo Semdegi Sefaseb-ebi:

1 2 3 4 5 6 7 8 9 10

8 7 19 18 10 11 18 17 17 13

11 12 13 14 15 16 17 18 19 20

14 18 15 16 14 16 11 20 18 20

ganmarteT qvemoT CamoTvlili cnebebi da gamoTvaleT: variaciuli

mwkrivi, moda, mediana, sixSire, dagrovili sixSire, procenti, dagr-

ovili procenti, saSualo, dispersia, saSualo kvadratuli gadaxra,dispersiis Sefaseba, saSualo kvadratuli gadaxris Sefaseba, ricxv

16-is rangi, misi procentuli rangi, mesame kvartili, meoTxe decili,

45 procentili, variaciis koeficienti.

9. wina amocanis monacemebi daajgufeT 5 intervalad. aageT his-

tograma, poligoni, daadgineT aris Tu ara mocemuli ganawileba unim-

odalri.

10. wamlis moqmedebis efeqturobis Sesamowmeblad Catarda 20

cda ( X wamlis dozaa, xolo Y avadmyofobis maxasiaTebeli):

X 8 6 5 1 0 2 4 3 8 10

Y 13 15 16 21 23 19 17 17 14 12

X 9 9 5 1 3 2 6 4 8 5

Y 13 12 18 20 18 17 14 16 13 18

aageT gabnevis diagrama, ipoveT korelaciis koeficienti, misi saSual-

ebiT daadgineT rogor kavSirSi imyofebian es cvladebi. SeadgineT re-gresiis wrfe da misi saSualebiT gaakeTeT prognozi: risi toli iq-

neba avadmyofobis maxasiaTebeli Tu wamlis doza iqneba 7-s toli.

11. leqciebze aqtiurobasa da sagnis warmatebiT aTvisebis Sesam-

owmeblad Catarda 20 cda ( X aqtiurobaa, xolo Y warmatebuleba):

X 2 16 18 10 8 6 9 13 15 12

Y 4 17 17 11 7 5 11 12 13 13

X 17 16 6 4 11 14 9 17 14 12

Y 16 17 4 5 10 12 11 15 15 13

44

aageT gabnevis diagrama, ipoveT korelaciis koeficienti, misi saSuaeb-

iT daadgineT rogor kavSirSi imyofebian es cvladebi. SeadgineT regr-

esiis wrfe da misi saSualebiT gaakeTeT prognozi: risi toli iqneba

warmatebuleba Tu aqtivoba iqneba 20-s toli.

II. albaTobis Teoria

l e q c i a 3.

Tavi 4. albaTobis Teoriis elementebi.

4.1. SemTxveviTi eqsperimenti, albaTuri sivrce, xdomiloba,

SemTxveviTi sidide.

cdas (eqsperiments), romlis Sedegis winaswar calsaxad gansaz-

Rvra SeuZlebelia, albaTobis TeoriaSi SemTxveviT eqsperiments uwod-eben. SemTxveviT eqsperimentTan dakavSirebul nebismier faqts, an mov-

lenas SemTxveviTi xdomiloba an ubralod, xdomiloba ewodeba. magal-

iTad, ra SeiZleba iTqvas SemTxveviT arCeul ojaxSi bavSvebis raoden-

obaze? SeiZleba Tu ara winaswar calsaxad imis Tqma, rom am ojaxSi

ori bavSvia? albaT, ara. ase, rom faqti imis Sesaxeb, rom SemTxveviTarCeul ojaxSi iqneba ori bavSvi warmoadgens xdomilobas, dakavSireb-

uls SemTxveviT sididesTan – (SemTxveviT arCeul) ojaxSi bavSvTa ra-

odenobasTan, xolo SemTxveviT eqsperiments am SemTxvevaSi warmoadg-ens dakvirveba am sidideze.

imis Tqma, rom Catarda SemTxveviTi eqsperimenti, gulisxmobs,

rom Cven principulad SegviZlia am eqsperimentis yvela SesaZlo Sede-

gze laparaki. amitom albaTobis TeoriaSi ama Tu im eqsperimentis yve-

la SesaZlo Sedegi ganixileba rogorc eqsperimentis calkeul SesaZ-

lo SedegTa erToblioba, rasac maTematikaSi simravles eZaxian. am sim-

ravles, Sesabamis eqsperimentTan dakavSirebul albaTur sivrces uwo-deben, xolo mis calkeul elements – elementarul xdomilobas. alb-

aTuri sivrce berZnuli didi “omega” asoTi aRiniSneba, xolo ele-

mentaruli xdomilobebi ki patara “omega” asoTi. is faqti, rom Ca-

tarda SemTxveviTi eqsperimenti, mokled ase gamoiTqmeba: mocemuliaalbaTuri sivrce = {1,2, 3, . . .}.

SevniSnoT, rom albaTuri sivrcis elementTa raodenoba SeiZle-

ba iyos sasrulic da usasruloc. sasrul elementian albaTur sivrc-

eze ityvian, rom is diskretulia, xolo elementTa usasrulo raode-

nobis SemTxvevaSi, albaTuri sivrce SeiZleba iyos diskretulic da

45

uwyvetic (gavixsenoT ricxviTi monacemebis tipebi). Tu laparakia gar-kveuli movlenis moxdenaTa raodenobaze, maSin laparakoben diskret-ul albaTur sivrceze, xolo Tu eqsperimentSi laparakia raime sidi-

dis gazomvaze, romelsac SeuZlia mniSvnelobebis miReba uwyvet skala-

ze (dro, sigrZe, wona, sisxlis wneva da a.S.), maSin Sesabamis albaTur

sivrcesac uwyvets uwodeben.xdomilobebi, romlebic gansazRvris Tanaxmad, albaTur sivrces-

Tan dakavSirebul faqtebs an movlenebs vuwodeT, faqtiurad, maTemat-

ikuri terminologiiT rom vTqvaT, albaTuri sivrcis nawilebs, anu

qvesimravleebs warmoadgenen da maT didi laTinuri asoebiT aRniSnaven.

magaliTad, xdomiloba “SemTxveviT arCeul ojaxSi ori bavSvia”, mokl-ed ase SegviZlia CavweroT: E = {X = 2}, sadac X SemTxveviTi sidideaRniSnavs bavSvTa raodenobas SemTxveviT amorCeul ojaxSi. aseve, xdo-

miloba “SemTxveviT amorCeuli avadmyofis sistoluri wneva naklebia

130.5 mm vwy.sv.-ze” SegviZlia ase CavweroT: E = {Y < 130.5}, sadac YSemTxveviTi sidide aRniSnavs SemTxveviT amorCeuli avadmyofis sisto-

luri wnevis sidides da a.S.

4.2. moqmedebebi xdomilobebze.

or E1 da E2 xdomilobas uTavsebadi (an urTierTgamomricxavi)ewodeba, Tu SeuZlebelia maTi erTdroulad moxdena, anu maTematikur-ad rom vTqvaT, E1 da E2 xdomilobebi, rogorc obieqtebis raRac simr-

avleebi, ar Seicavs arc erT saerTo obieqts. am faqts mokled ase

weren: E1 E2 = , sadac aris SeuZlebeli xdomilobis (anu iseTixdomilobis, romelic arcerT elementarul xdomilobas ar Seicavs)

simboluri aRniSvna. magaliTad, SeuZlebelia erTdroulad erTidaimave

avadmyofis sistoluri wneva naklebi iyos 130.5 mm vwy.sv.-ze da meti

iyos 154.5 mm vwy.sv.-ze, anu{Y 130.5}{Y > 154.5}= .

TanakveTis simboloTi aRniSnulia xdomilobaTa erTdroulad

moxdenis faqti. roca laparakia ori (an meti) xdomilobidan erTi ma-

inc xdomilobis moxdenaze, am faqts gaerTianebis simboloTi aRniS-naven.

magaliTad, xdomiloba “SemTxveviT arCeul ojaxSi erTi an ori

bavSvia”, Caiwereba ase: {X = 1}{X = 2}.SevniSnoT, rom xdomilobaTa gaerTianeba (erTis mainc moxdena)

ar gamoricxavs maT TanakveTas, am dros SesaZlebelia orives erTdro-

ulad moxdenac. magaliTad, xdomilobebis A = {avadmyofis sistoluri

wneva naklebia 130.5 mm vwy.sv.-s} da B = {avadmyofis sistoluri wneva

meti an tolia 106.5 mm vwy.sv.-ze da naklebia 154.5 mm vwy.sv.-ze} gaer-Tianeba, niSnavs xdomilobis C = {avadmyofis sistoluri wneva nakle-

46

bia 154.5 mm vwy.sv.-s} ganxorcielebas, romelic Seicavs mocemuli orixdomilobis TanakveTasac D = {avadmyofis sistoluri wneva meti an

tolia 106.5 mm vwy.sv.-ze da naklebia 154.5 mm vwy.sv.-ze}, anu:AB ={Y < 130.5}{106.5 Y < 154.5}={Y < 154.5}= C D =={106.5 Y < 130.5}={Y < 130.5}{106.5 Y < 154.5}= AB.simbolos, romelic Cven aq gamoviyeneT, (), e.w. CarTvis sim-

bolos uwodeben da is gamoiyeneba imis saCveneblad, Tu romeli xdom-

ilobaa romlis nawili anu qvesimravle. Cvens SemTxvevaSi, D ={106.5Y<130.5} xdomiloba nawilia C={Y<154.5} xdomilobisa, anu

{106.5Y<130.5}{Y<154.5}. gavixsenoT, rom sazogadod, nebismieri Exdomiloba warmoadgens -s qvesimravles, anu E .

SemoviRoT E xdomilobis damatebiTi xdomilobis (sawinaaRmdegoxdomilobis) cnebac. Tu eqsperimentis Sedegad ar moxda sasurveli Exdomiloba, maSin amboben, rom momxdara misi damatebiTi EC xdomiloba.urTierTdamatebiT xdomilobebs is Tviseba aqvs, rom

E EC = da E EC = .xSirad xdomilobebis gaerTianebas xdomilobaTa jams uwodeben

da weren E1 + E2-s , E1 E2 -is nacvlad, anu E1 E2 = E1 + E2.

aseve, xdomilobaTa TanakveTis nacvlad xSirad ixmareba xdomi-lobaTa namravli da weren E1 E2 = E1 E2.

xdomilobaTa SekrebasTan erTad SeiZleba visaubroT xdomilob-

aTa sxvaobazec. ori E1 da E2 xdomilobis sxvaoba ewodeba xdomilobas,

romelic gulisxmobs pirveli xdomilobis moxdenasa da meoris ar mo-

xdenas erTdroulad, anu

E1 \ E2 = E1 E2C = E1 E2

C.

SevniSnoT, rom \ E = EC = EC, anu E xdomilobis damatebi-

Ti xdomiloba SegveZlo gangvesazRvra, rogorc da E xdomilobebis

sxvaoba: EC = \ E. aqedan kerZod davaskvniT, rom C = \ = da C

= . ukanaskneli toloba gveubneba, rom SeuZlebeli xdomilobis sawi-

naaRmdego (damatebiT) xdomilobas warmoadgens albaTuri sivrce da

aqedan gamomdinare, rogorc xdomilobas, mas aucilebel xdomilobasuwodeben.

CamovTvaloT zemoT Semotanili xdomilobaTa operaciebis Tvis-ebebi, romlebic dagvWirdeba momavalSi:

1) E1 E2 E1 da E1 E2 E2;2) E1 E2 E1 da E1 E2 E2;3) ( EC )C = E;4) E1

C E2C = (E1 E2)

C ;5) E1

C E2C = (E1 E2)

C ;

47

6) E1 \ E2 = E1 \ (E1 E2);7) E1 E2 = (E1 \ E2) + (E1 E2) + (E2 \ E1)8) (E1 E2) E3 = (E1 E3) (E2 E3);9) (E1 E2) E3 = (E1 E3) (E2 E3).moxerxebulia operaciebi xdomilobebze warmodgenili iqnas di-

agramebis, e.w. venis diagramebis saxiT.

4.3. xdomilobis albaToba, albaTobaTa Tvisebebi.

ama Tu im xdomilobis albaToba ewodeba am xdomilobis fardo-

biT sixSires cdebis usasrulod didi raodenobis dros.

sinamdvileSi, cxadia, rom veravin Caatarebs usasrulo raoden-

obis dakvirvebebs ama Tu im movlenaze. amitom, rogorc wesi, xdomil-

obis albaTobad iReben mis Sefasebas – xdomilobis fardobiT sixSir-

es cdebis sakmarisad didi raodenobisas. xSir SemTxvevaSi ki, iqcevian

ase: elementarul (an raime martivi tipis) xdomilobebs garkveuli

mosazrebebis (vTqvaT, simetrulobis) gamo miaweren garkveul mniSvne-lobebs da maTi saSualebiT iTvlian SedarebiT rTul xdomilobaTa

albaTobebs.

magaliTi 4.1. cxrilSi moyvanilia aSS-Si Catarebuli gamokvlev-

is Sedegebi axalSobilTa sqesis Sesaxeb:

kalendaruli

dro

dabadebuli

biWebis raod.

axalSobilTa

saerTo raod.

fardobiTi sixSire

1965 1 927 054 3 760 358 0.51247

48

1965-1969 9 219 202 17 989 361 0.512481965-1974 17 857 857 34 832 051 0.51268

albaT Znelia, am monacemebis mixedviT, eWvis Setana imaSi, rom

xdomilobis “aSS-Si axalSobili iqneba biWi” albaToba (measedamde si-

zustiT) tolia 0.51-isa (da ara 0.50-isa, rogorc es bevrs SeiZleba

moeCvenos “simetrulobis” gamo).

CamovayaliboT xdomilobaTa albaTobebis zogierTi ZiriTadi

Tviseba. daviwyoT imiT, rom rogorc fardobiT sixSireTa (anu [0,1] in-tervalSi moqceul sidideTa) zRvruli SemTxveva, nebismieri xdomil-

obis albaTobac moTavsebulia amave intervalSi, anu nebismieri E xdo-

milobisaTvis: 0 P(E) 1, sadac P(E) aRniSnavs E xdomilobis albaTo-

bas.

cxadia, rom SeuZlebeli xdomilobis fardobiTi sixSire nulis

tolia cdaTa nebismieri raodenobis dros. ase, rom P{}=0. aucilebe-li xdomiloba yovel eqsperimentSi xorcieldeba, anu misi fardobiTi

sixSire cdebis nebismieri raodenobisaTvis 1-is tolia da maSasadame,

P{ } = 1.davuSvaT, rom E1 E2, anu E1 nawilia E2 xdomilobis (am SemTxv-

evaSi amboben, rom E1 iwvevs E2-s). vTqvaT, n cdaSi maTi moxdenaTa rao-denobebia Sesabamisad n(E1) da n(E2). maSin cxadia, rom n(E1) n(E2).Sesabamisad, n(E1)/n n(E2)/n, anu maTi fardobiTi sixSireebic akmayofi-lebs imave utolobas fn(E1) fn(E2) da maSasadame,

P(E1) P(E2), roca E1 E2.

am SemTxvevaSi amboben, rom albaTobas aqvs zrdadobis Tviseba.davuSvaT axla, rom yovel cdaSi SeuZlebelia E1 da E2 xdomil-

obis erTdroulad moxdena, anu E1 da E2 xdomilobebi uTavsebadia. maS-in cxadia, rom n cdaSi maTi jamuri xdomilobis moxdenaTa raodenoba

tolia calkeuli xdomilobis moxdenaTa raodenobebis jamisa, anu

n(E1+E2)= n(E1) + n(E2). amitom maTi fardobiTi sixSireebi akmayofilebs

pirobas:fn(E1 + E2) = n(E1 + E2)/n = (n(E1) + n(E2))/n =

=n(E1)/n + n(E2)/n = fn(E1) + fn(E2),anu jamuri xdomilobis fardobiTi sixSire Sesakreb xdomilobaTa fa-

rdobiTi sixSireebis jamis tolia. radgan es Tanafardoba sworia neb-

ismierad didi n-saTvis, is sworia iqneba zRvruli SemTxvevisTvisac,

anu albaTobebisaTvisac da Cven vRebulobT e.w. albaTobaTa Sekrebiswess: Tu E1 da E2 xdomilobebi uTavsebadia, maSin:

P(E1 + E2) = P(E1) + P(E2). (4.1)

unda iTqvas, rom Cvens mier danaxuli es umartivesi Tvisebebi

warmoadgens albaTobis Teoriis aqsiomatur safuZvels, anu yvela sxva

49

Tviseba maTgan gamomdinareobs. maTi saSualebiT qvemoT Cven gamoviyvanTalbaTobaTa sxva Tvisebebs, romlebic Cven momavalSi dagvWirdeba.

albaTobaTa Sekrebis (4.1) wesidan Tavis mxriv gamomdinareobs

albaTobaTa gamoklebis wesi:

Tu raime A xdomiloba iwvevs B xdomilobas, anu A B, maSinP(B \ A) = P(B) - P(A), (4.2)

xolo nebismieri E1 da E2 xdomilobisaTvis

P(E1 \ E2) = P(E1) - P(E1E2). (4.3)ukanaskneli toloba gamomdinareobs xdomilobaTa me-6 da pirveli

Tvisebebidan.

SevniSnoT, rom induqciiT advilad davrwmundebiT, rom xdomil-

obaTa Sekrebis wesi samarTliania ara mxolod ori uTavsebadi xdomi-

lobisaTvis, aramed, wyvilwyvilad uTavsebad xdomilobaTa nebismieri

sasruli raodenobisaTvis, anu

Tu Ei Ej = , nebismieri i j-saTvis, i , j = 1,2,…, k, maSinP(E1 + E2 + … + Ek) = P(E1) + P(E2) + … + P(Ek). (4.4)

albaTobaTa Sekrebis am wess albaTobaTa Sekrebis ganzogadeb-uli wesi ewodeba.

sailustraciod gavixsenoT 2.3 magaliTi. CavTvaloT xdomilob-

aTa fardobiTi sixSireebi albaTobebad da gamovTvaloT albaToba imi-

sa, rom avadmyofis sistoluri wneva naklebia 154.5 mm vwy.sv.-ze, anuCvens aRniSvnebSi, C = {Y < 154.5}. es xdomiloba SegviZlia CavweroT

eqvsi wyvil-wyvilad uTavsebadi xdomilobis jamis saxiT:

{Y<154.5}={82.5 Y < 94.5}+{94.5 Y <106.5}+{106.5 Y <118.5}++{118.5 Y < 130.5}+{130.5 Y <142.5}+{142.5 Y <154.5}.

amitomP{C}=P{Y <154.5} = 6/120+20/120+17/120+30/120 +25/120+13/120 =

=111/120.xdomilobaTa Sekrebisa da nebismier xdomilobaTa albaTobebis

gamoklebis wesiT me-7 Tvisebidan SeiZleba gamoviyvanoT xdomilobaTa

Sekrebis wesi nebismieri ori xdomilobisaTvis. marTlac,

P(E1 E2) = P(E1 \ E2) + P(E1E2) + P(E2 \ E1) == P(E1) - P(E1E2) + P(E1E2) + P(E2) - P(E1E2) = P(E1) + P(E2) - P(E1E2),

anu sabolood, albaTobaTa Sekrebis wesi nebismieri ori xdomilob-

isaTvis SemdegSi mdgomareobs:

nebismieri ori xdomilobis jamis albaToba tolia am xdomil-

obaTa albaTobebis jams gamoklebuli maTi TanakveTis (saerTo nawil-

is) albaToba, anu

P(E1 E2) = P(E1) + P(E2) - P(E1E2). (4.5)sailustraciod gavixsenoT avadmyofebis sistoluri wnevis 2.3

magaliTi. rogorc zemoT vnaxeT, P{C}=P{Y<154.5}=111/120. (4.5) form-uliT igive albaToba SegviZlia gamovTvaloT ase:

50

P{C}= P{Y < 154.5} = P{{Y <130.5}{106.5<Y 154.5}}= P{Y 130.5}++P{106.5 Y <154.5}- P{{Y 130.5}{106.5 <Y 154.5}}= P{Y 130.5}+

+ P{106.5 Y <154.5}- P{106.5 Y < 130.5}= 73/120+85/120-47/120=111/120.

4.4. pirobiTi albaToba. albaTobaTa gamravlebis wesi.

xdomilobaTa damoukidebloba.

A xdomilobis pirobiTi albaToba im pirobiT, rom moxda Bxdomiloba, P{A | B} simboloTi aRniSneba da ase gansazRvreba:

P{A | B}= P{AB}/ P{B}, (4.6)

anu A xdomilobis pirobiTi albaTobis gamosaTvlelad im pirobiT,

rom moxda B xdomiloba, saWiroa A xdomilobis im nawilis albaTo-

bis Sefardeba B xdomilobis albaTobasTan, romelic B xdomilobaSi

Sedis.

gadavweroT (4.6) formula Semdegi saxiT:

P{AB} = P{A | B} P{B}. (4.7)

(4.7) formuliT gamosaxul tolobas albaTobaTa gamravlebiswesi ewodeba.

gavixsenoT 2.2 magaliTi. CavTvaloT miRebuli fardobiTi sixSi-

reebi albaTobebad da gamovTvaloT pirobiTi albaToba imisa, rom

“ojaxSi ori bavSvia” (aRvniSnoT es xdomiloba A-Ti) im pirobiT, rom“ojaxSi erTi mainc bavSvia” (aRvniSnoT es xdomiloba B-Ti). gavixsen-oT, rom orbavSviani ojaxebis raodenoba iyo 23, xolo im ojaxebis

raodenoba, romelSic erTi mainc bavSvia, tolia 87-15=72, anu ojaxeb-is saerTo raodenobas gamoklebuli ubavSvo ojaxebis raodenoba. rad-

gan am SemTxvevaSi AB = A, amitom (4.6) formulis Tanaxmad, miviRebT:

P{A | B}= P{AB}/ P{B}= P{A}/ P{B} = (23/87)/(72/87) = 23/72.analogiurad, gavixsenoT, rom avadmyofTa sistoluri wnevis

Sesaxeb magaliTSi A-Ti aRniSnuli iyo xdomiloba: “avadmyofis sist-

oluri wneva ar aRemateba 130.5 mm vwy.sv.-s”, xolo B-Ti ki xdomilo-

ba: “avadmyofis sistoluri wneva moTavsebulia 106.5-sa da 154.5 mmvwy.sv.-s Soris”. amitom (4.6) formulis mixedviT, gveqneba:

P{A | B}= P{AB}/ P{B}= P{106.5 Y < 130.5}/ P{106.5 Y < 154.5}==(47/120)/(85/120) = 47/85.

SevniSnoT, rom ukanasknel SemTxvevaSi AB A, gansxvavebiT winaSemTxvevisagan. SevadaroT A xdomilobis pirobiTi albaTobebi Sesabam-

is upirobo albaTobebs orive magaliTisaTvis. rogorc vnaxeT, pirvel

SemTxvevaSi, P{A|B} = 23/72, xolo A xdomilobis upirobo albaTobaa

P{A} = 23/87, anu Cven vxedavT, rom P{A | B}> P{A};meore SemTxvevaSi, P{A|B}=47/85 da P{A}=P{Y120}=73/120 da

maSasadame, P{A|B}<P{A}. sabolood, erTi magaliTisaTvis gvaqvs, rom

51

P{A|B}> P{A}, meoresTvis ki, piriqiT, P{A|B}<P{A}. aqedan gasagebia,rom principSi albaT, SesaZlebelia iseTi ori xdomilobis mofiqrebac

(garkveuli eqsperimentis dros), rom adgili eqnes tolobas: P{A|B}==P{A}. SevniSnoT, rom aseT SemTxvevaSi, P{AB} = P{A} P{B} da

P{A|BC}= P{ABC}/P{BC}= (P{A}-P{AB})/(1-P{B})=P{A}(1-P{B})/(1-P{B})= P{A}, anu sabolood,

P{A|B}= P{A|BC}= P{A}. (4.8)

(4.8) formula niSnavs, rom: A xdomilobis albaToba araa dam-

okidebuli imaze, moxda Tu ara B xdomiloba.

aseT SemTxvevaSi bunebrivia, amboben, rom A xdomiloba albaTur-ad damoukidebelia, an ubralod, damoukidebelia B xdomilobisagan.

(4.7) Tanafardobidan gamomdinareobs albaTobaTa gamravlebiswesi damoukidebeli xdomilobebisaTvis:

P{AB} = P{A} P{B}. (4.9)

(4.9) formulis simetrulobis gamo davaskvniT, rom Tu A xdomiloba

damoukidebelia B xdomilobisagan, maSin B xdomilobac damoukidebe-

lia A xdomilobisagan.

SevniSnoT, rom albaTobaTa Sekrebis ganzogadebuli wesis msgav-

sad, SesaZlebelia albaTobaTa namravlis wesis Semdegi ganzogadeba:

P{E1 E2 Ek }=P{E1}P{ E2| E1}P{ Ek | E1 E2 Ek-1 }.amocana. davuSvaT, rom Sesamowmebeli jgufis 1% avadmyofia,

xolo danarCeni 99% ki janmrTelia. adamianebis SerCeva xdeba SemTxv-

eviT da amitom

( ) 1% 0.01P avadmyofi da ( ) 99% 0.99P janmrTeli .

vigulisxmoT, rom im SemTxvevaSi, roca testireba utardeba adamians,

romelsac ara aqvs avadmyofoba, maSin 1%-ia albaToba imisa, rom mivi-RoT mcdari dadebiTi Sedegi, e.i.

%1)( janmrTeli|dadebiTiP da

%99)( janmrTeli|uaryofiTiP .

dabolos, davuSvaT, rom im SemTxvevaSi, roca testireba utardeba ava-

dmyof adamians, maSin 1%-ia albaToba imisa, rom miviRoT mcdari uar-

yofiTi Sedegi, e.i.

%1)( avadmyofi|uaryofiTiP da %99)( avadmyofi|dadebiTiP .

gamoTvaleT albaToba imisa, rom: a). adamiani janmrTelia, xolo test-

ma aCvena uaryofiTi Sedegi; b). adamiani avadmyofia, xolo testma aCve-

na dadebiTi Sedegi; g). adamiani janmrTelia, xolo testma aCvena dad-

ebiTi Sedegi; g). adamiani avadmyofia, xolo testma aCvena uaryofiTi

Sedegi.

ganvixiloT realuri situacia, romelic gviCvenebs erTi Sexed-

viT moulodnel gansxvavebas )|( BAP da )|( ABP pirobiT albaTob-

52

ebs Soris. imisaTvis, rom gamovavlinoT seriozuli avadmyofobis mqo-ne adamianebi adreul stadiaze, xdeba adamianebis didi jgufis testi-

reba. miuxedavad winaswari Semowmebis sargeblobisa, am midgomas gaaC-

nia uaryofiTi mxare: Tu adamians sinamdvileSi ar gaaCnia avadmyofoba

da sawyisma testma aCvena dadebiTi Sedegi (daudgina avadmyofoba), is

iqneba stresul mdgomareobaSi (rac Tavis mxriv uaryofiTad moqmedebs

mis cxovrebaze) sanam ufro warmatebuli testi ar aCvenebs, rom is

janmrTelia. am problemis mniSvneloba SesaZlebelia kargad gavigoT

pirobiTi albaTobebis terminebSi.

zemoT moyvanili amocanis pirobebSi gamovTvaloT albaToba imi-

sa, rom testi aCvenebs dadebiT Sedegs. sruli albaTobis formulisTanaxmad:

( ) ( ) ( )P P P dadebiTi janmrTeli dadebiTi | janmrTeli

( ) ( )P P avadmyofi dadebiTi | avadmyofi

0.99 0.01 0.01 0.99 0.0198 .

rogorc cnobilia, amocanis pirobebSi

%99)( avadmyofi|dadebiTiP .

gamovTvaloT axla Sebrunebuli pirobiTi albaToba, risTvisac

visargebloT pirobiTi albaTobis ganmartebiTa da namravlis albaTob-

is formulebiT. maSin zemoT miRebuli( ) 0.0198 1.98%P dadebiTi

Sedegis Tanaxmad:

(dadebiTi)PP

dadebiTiavadmyofidadebiTi|avadmyofi

P

PP

)()(

( ( ) 1% 99%50%

1.98%

P P

avadmyofi)P dadebiTi | avadmyofi

PP1.98%.

rogorc vxedavT, pirobiTi albaToba imisa rom testi mogvcemsdadebiT Sedegs, pirobaSi rom adamiani avadmyofia tolia 99%-is, maS-

in rodesac pirobiTi albaToba imisa rom adamiani avadmyofia, pirobaSi

rom testma mogvca dadebiTi Sedegi aris mxolod 50%. aq SerCeuli

monacemebis SemTxvevaSi ukanaskneli Sedegi SeiZleba CaiTvalos miuReb-

elad: naxevari adamianebis, romelTa testirebam aCvena dadebiTi Sede-

gi, faqtiurad aris mcdari dadebiTi.

4.5. sruli albaTobisa da baiesis formulebi.

albaTobaTa Sekrebis ganzogadebul wesSi laparaki iyo k cal

wyvil-wyvilad uTavsebad xdomilobaze anu iseT xdomilobebze, rome-

lTaTvisac Ei Ej = , nebismieri i j-saTvis, i , j = 1,2,…, k. davuSv-aT, rom garda zemoT aRniSnuli Tvisebisa, es xdomilobebi “avseben”

53

mTel albaTur sivrces, anu E1 + E2 + … + Ek = . maSin xdomilobaTaaseT sistemas xdomilobaTa srul sistemas uwodeben.

sruli albaTobis formula gamoiyeneba raime rTuli C xdomi-lobis albaTobis gamosaTvlelad da mas Semdegi saxe aqvs:

P{C}=

k

iii EPECP

1

}{}|{ =P{C|E1}P{E1}+P{C|E2}P{E2}+

+…+P{C|Ek}P{Ek}, (4.10)

sadac E1, E2 , … , Ek warmoadgens xdomilobaTa srul sistemas.

am formulis gamoyenebis sailustraciod moviyvanoT Semdegi

magaliTi 4.2. gamokvlevebiT dadgenilia, rom 60-64ww. asakobr-

ivi jgufis adamianebisaTvis momdevno xuT weliwadSi kataraqtiT daa-

vadebis albaTobaa 0.024, 65-69ww. asakobriv jgufSi myofTaTvis –0.046, 70-75ww. asakobrivi jgufisaTvis –0.088 da 75 wels gadacile-

bulTaTvis – 0.153. 5000 kacian populaciaSi, romlebic samoc wels

gadacilebuli arian, 45%-is asaki aRmoCnda 60-64ww. asakobriv inter-

valSi, 28%-isa – 65-69ww. intervalSi, 20%-isa –70-74ww. interval-

Si da danarCenebi 75 wels gadacilebulni aRmoCndnen. ras udris alb-

aToba imisa, rom xuTi wlis Semdeg SemTxveviT amorCeuli pirovneba

aRmoCndeba kataraqtiT daavadebuli?

amoxsna: SemoviRoT xdomilobebi:

C = {SemTxveviT amorCeuli pirovneba kataraqtiTaa daavadebuli};E1={amorCeuli pirovnebis asaki moTavsebulia 60-64ww. intervalSi};E2={amorCeuli pirovnebis asaki moTavsebulia 64-69ww. intervalSi};E3={amorCeuli pirovnebis asaki moTavsebulia 70-74ww. intervalSi};E4={amorCeuli pirovnebis asaki metia 75 welze}.

radgan erTi, SemTxveviT amorCeuli pirovnebis asaki ar SeiZle-

ba moxvdes erTdroulad or TanaukveT asakobriv intervalSi, amitom

Ei Ej = , nebismieri i j-saTvis, i , j = 1,2,3,4. garda amisa, radganyvela pirovneba samoci wlis asaks gadacilebulia, nebismieri pirovne-

ba aucileblad moxvdeba romelime zemoT CamoTvlil asakobriv inter-

valSi; amitom E1 + E2 + E3+ E4 = , anu E1, E2 , E3, E4 xdomilobebi Sead-

genen xdomilobaTa srul sistemas. pirobis Tanaxmad,

P{C|E1}= 0.024, P{C|E2}= 0.046, P{C|E3}= 0.088, P{C|E4}= 0.153da P{E1}= 0.45, P{E1}= 0.28, P{E1}= 0.2, P{E1}= 0.07. amitom (4.10)formulis Tanaxmad miviRebT:

P{C}=

4

1

}{}|{i

ii EPECP = 0.0240.45 + 0.0460.28 +

+0.0880.2 + 0.1530.07 = 0.052.Sesabamisad, SegviZlia davaskvnaT, rom populaciis 5.2%, anu

260 adamiani, xuTi wlis Semdeg daavadebuli iqneba kataraqtiT.

54

davuSvaT axla, rom xuTi wlis Semdeg SemTxveviT amorCeulipirovneba aRmoCnda kataraqtiT daavadebuli. sainteresoa, ras udris

albaToba imisa, rom es pirovneba xuTi wlis win 60-64ww. asakobrivi

jgufis (an sxva romelime jgufis) warmomadgeneli iyo? SevniSnoT,

rom zemoT ganxilulisagan gansxvavebiT, cdis (SemTxveviT amorCeuli

pirovneba iqneba kataraqtiT daavadebuli) Sedegi cnobilia da gvainte-

resebs aseTi “hipoTezis” albaToba: es pirovneba xuTi wlis win 60-

64ww. asakobrivi jgufis warmomadgeneli iyo. SekiTxvis ase dasma sa-

interesoa imitom, rom SevadaroT hipoTezis aposterioruli (cdis

Semdegi) albaToba mis apriorul (cdis Catarebamde) albaTobas, rome-lic rogorc gvaxsovs, 0.45-is tolia (anu yvelaze didia apriorulalbaTobebs Soris). amis saSualebas iZleva baiesis formula, romels-

ac Semdegi saxe aqvs:

kjEPECP

EPECPCEP

k

iii

jjj ,...,2,1,

}{}|{

}{}|{}|{

1

. (4.11)

ganxilul magaliTSi, (4.11)-is marjvena mxareSi mdgomi gamosax-

ulebis mniSvneli rogorc vnaxeT, tolia 0.052-is. amitomP{E1|C}= (0.0240.45)/0.052 0.20,P{E2|C}= (0.0460.28)/0.052 0.25,P{E3|C}= (0.0880.20)/0.052 0.34,P{E4|C}= (0.1530.07)/0.052 0.21.

rogorc vxedavT, mas mere rac cnobilia, rom SemTxveviT amorC-

euli pirovneba daavadebulia kataraqtiT, yvelaze meti “Sansia” isxuTi wlis win yofiliyo 70-74ww. asakobrivi jgufis warmomadgeneli.

amocanebi1. ganvixiloT ojaxi, romelSic aris deda, mama da ori Svili.

SemoviRoT Semdegi xdomilobebi: E1 = {dedas gripi aqvs}, E2 = {mamasgripi aqvs}, E3 = {pirvel Svils gripi aqvs}, E4 = {meore Svils gripi

aqvs}, B = {erT Svils mainc gripi aqvs}, C={erT mSobels mainc gripi

aqvs}, D={erTs mainc gripi aqvs ojaxSi}.1.1. ras niSnavs xdomilobebi E1 E2, E1 E2 ;

1.2. aris Tu ara uTavsebadi xdomilobebi E3 da E4?

1.3. ras niSnavs xdomilobebi E3 B, E3 B ;1.4. gamosaxeT C xdomiloba E1, E2, E3 da E4 xdomilobebis saSualebiT.

1.5 gamosaxeT D xdomiloba B da C xdomilobebis saSualebiT.

1.6. ras niSnavs xdomilobebi E3C da E4

C ?

1.7. gamosaxeT DC xdomiloba B da C xdomilobebis saSualebiT.

55

2. ganvixiloT ojaxebi, sadac or-ori bavSvia. rogoria albaTobaimisa, rom ojaxSi orive bavSvi vaJia Tu cnobilia, rom: a). ufrosi

bavSvi – vaJia; b). erTi bavSvi mainc – vaJia?

3. yuTSi moTavsebulia ori moneta: 1A -- simetriuli moneta

gerbis mosvlis albaTobiT 1/2, da 2A -- arasimetriuli moneta gerb-

is mosvlis albaTobiT 1/3. SemTxveviT viRebT erT monetas da vagdebT.

davuSvaT, rom movida gerbi. rogoria albaToba imisa, rom amoRebulimoneta iyo simetriuli?

4. vTqvaT, Cven Casabarebeli gvaqvs gamocda da SegviZlia avirCi-

oT nebismieri sami gamomcdelidan. davuSvaT, CvenTvis cnobilia, rom

erTerTi sami gamomcdelidan (ucnobia romeli) -- “keTilia” da alba-

Toba imisa, rom masTan Caabaro gamocda tolia 0.4-is, xolo danarCeni

ori gamomcdeli “avia” da maTTan gamocdis Cabarebis albaToba tolia

0.1-is. Cven SemTxveviT avirCieT gamomcdeli da warmatebiT CavabareT

gamocda. rogoria labaToba imisa, rom Cven avirCieT “keTili” gamomc-

deli?

5. davuSvaT, rom gamomcdelTan, romelTanac warmatebiT Caiaragamocdam (ix. amocana 4) gamosacdelad rig-rigobiT mivida kidev ori

moswavle. jer gamocda ver Caabara meore moswavlem, Semdeg mivida me-

same da manac ver Caabara gamocda. am faqtis Semdeg romeli hipoTezaa

ufro dasajerebeli: es gamomcdeli “keTilia” Tu “avi”?

6. moyvaruli sinoptikosis Teoriis Tanaxmad Tu erT wels iyo

wyaldidoba, maSin albaToba imisa, rom igi ganmeordeba momdevno wels

aris 0.7, xolo Tu erT wels ar iyo wyaldidoba, maSin albaToba imi-

sa, rom igi ar iqneba momdevno wels aris 0.6. gasul wels wyaldido-

ba ar yofila. ipoveT albaToba imisa, rom wyald-idoba iqneba: a). mom-

devno sam weliwads zedized; b). zustad erTjer momdevno sami wlisganmavlobaSi.

7. albaToba imisa, rom zafxulis romelime dRes baTumSi iwvim-

ebs aris 0.2. albaToba imisa, rom dRis maqsimaluri temperatura wvi-

mian dRes 25 graduss gadaaWarbebs aris 0.3, xolo Tu wvima ar aris

– 0.6. vipovoT albaToba imisa, rom zafxulis SemTxveviT arCeul

dRes baTumSi iwvimebs, Tu cnobilia, rom am dRis maqsimaluri temper-

atura 25 gradusze metia.

8. hipertoniiT daavadebul avadmyofebs aZleven or sxvadasxva

wamals, romlebidanac pirvelis kuWnawlavze gvrdiTi zemoqmedebis

albaTobaa 0.1, xolo meorisa – 0.2. davuSvaT, rom es gverdiTi efeqt-ebi damoukidebelia erTmaneTisagan. ras udris albaToba imisa, rom av-

admyofs eqneba ukuCveneba am wamlebis erTdroulad miRebisas?

9. qimiur qarxanaSi momuSave 500 mamakacidan, romelTa asakia

50-dan 69 wlamdea, 35%-is asakia 50-54ww. asakobriv jgufSi, 30%-isa

56

55-59ww. asakobriv jgufSi, 20% -- 60-64ww. asakobriv jgufSi da da-narCeni - 65-69ww. asakobriv jgufSi. sikvdilianobebi am asakobrivi

jgufebisaTvis Sesabamisad Seadgens 0.9%, 1.4%, 2.2% da 3.3%-s. rog-

oria sikvdilianobis sidide am qarxanaSi?

10. axalSobili iTvleba dabalwonianad Tu misi wona ar aRema-

teba 2500 grams. garda amisa, axalSobilebi klasificirdebian kidev

fexmZimobis periodis sididiTac: 20 kviraze naklebi, 20-27 kvira, 28-

36 kvira da 36 kviraze meti. davuSvaT, rom fexmZimobis periodebis

Sesabamisi albaTobebia: 0.0004, 0.0059, 0.0855, 0.9082. cnobilia agr-

eTve dabalwonianobis Sesabamisi albaTobebi: 0.540, 0.813, 0.379 da

0.035 (magaliTad, albaToba 0.540 gviCvenebs albaTobas, imisa rom axa-lSobili iqneba dabalwonianad, Tu fexmZimobis periodi ar aRemateba

20 kviras).

10.1. gamoTvaleT albaToba imisa, rom SemTxveviT arCeuli axalSobili

dabalwoniani iqneba;

10.2. aCveneT, rom xdomilobebi “fexmZimobis periodi ar Remateba 27

kviras” da “SemTxveviT arCeuli axalSobili dabalwoniani iqneba” ar

aris damoukidebeli.

10.3. ras udris albaToba imisa, rom fexmZimobis periodi ar aRemateba

36 kviras, Tu axalSobili dabalwoniani aRmoCnda?

11. daavadeba taxemia SeiZleba ganviTardes orsulobis dros.taxemia ufro xSirad viTardeba diabetiT daavadebul qalebSi. cnobi-

lia, rom qalebis daaxloebiT 20%-s aqvs diabeti da am qalebis 25%-

s SeiZleba ganuviTardes taxemia. janmrTeli orsuli qalebis daaxlo-

ebiT 4.6%-Si fiqsirdeba taxemia. gamoTvaleT albaToba imisa, rom Sem-

TxveviT SerCeul orsul qals aqvs taxemia.

12. erT-erT kunZulze aRmoCenil iqna mcenare hebe-s ori saxeo-

ba: makrouta da atkinsoni . am kunZulze makrouta Seadgens hebe-s sa-

xeobis 70%-s. albaToba imisa, rom makrouta iqneba sworad identifi-

cirebuli aris 60% da, rom atkinsoni ar iqneba sworad identifici-

rebuli aris 6%. gamoTvaleT albaToba imisa, rom SemTxveviT arCeulihebe aris sworad identificirebuli.

13. Sizofrenia kacebs ufro xSirad emarTebaT vidre qalebs. Si-

zofreniiT daavadebis albaToba kacebSi Seadgens 0.0003-s, xolo qal-

ebSi ki 0.0002-s. garkveuli wamali kacebisaTvis ufro efeqturia (da-

debiTi efeqtis albaTobiT 0.7), vidre qalebisaTvis (aq dadebiTi efeq-

tis albaTobaa 0.4). gamoTvaleT albaToba imisa, rom SemTxveviT amor-

Ceul pacientze wamali dadebiTad imoqmedebs.

57

l e q c i a 4.

Tavi 5. diskretuli SemTxveviTi sidideebi. ZiriTadi

albaTuri ganawilebebi.

5.1. SemTxveviTi sidideebi da maTi ricxviTi maxasiaTeblebi.

wina leqciaSi Cven ukve gavecaniT SemTxveviTi sididis magaliT-

ebs: SemTxveviT arCeul ojaxSi bavSvTa raodenoba, axalSobilebis wo-

na, avadmyofTa sisxlis wneva da a. S. magram iq ar mogviyvania sazoga-

dod SemTxveviTi sididis gansazRvra. es leqcia swored SemTxveviTi

sidideebs da maTTan dakavSirebul albaTur ganawilebebs daeTmoba.

gansazRvra 5.1. SemTxveviTi sidide es aris ricxviT mniSvnelob-

ebiani sidide, romelic Rebulobs sxvadasxva ricxviT mniSvnelobebs,

Sesabamisi albaTobebiT.

ganasxvaveben SemTxveviTi sidideebis or ZiriTad tips: diskret-ul da uwyvet SemTxveviT sidideebs.

gansazRvra 5.2. diskretulia SemTxveviTi sidide, romlis SesaZ-

lo mniSvnelobaTa simravlec diskretulia, anu sasruli an Tvladia.

es gansazRvra gulisxmobs, rom Cven SegviZlia CamovTvaloT Sem-

TxveviTi sididis yvela SesaZlo mniSvneloba. magaliTad, ojaxSi bavS-

vTa raodenoba SeiZleba iyos 0, 1, 2, …10, 11,…,20 (SeiZleba metic, mag-

ram yovel SemTxvevaSi, raRac sasruli ricxvi).

sxva SemTxvevaSi, SeiZleba laparaki iyos diskretul SemTxveviT

sidideze, romelis SesaZlo mniSvnelobaTa raodenoba SeiZleba iyos

ragind didi. magaliTad, otiti (Otitis media) aris Sua yuris anTeba,romelic Zalzed xSiria or wlamde asakis bavSvebSi da Tu X aRniSn-avs am daavadebiT daavadebaTa SemTxvevebis raodenobas, maSin gasagebia,

rom misi SesaZlo mniSvnelobebia 0,1,2,….

rodis vityviT, rom mocemulia diskretuli X SemTxveviTi si-dide? SemTxveviTi sididis gansazRvridan gamomdinare, amisaTvis araa

sakmarisi mxolod misi mniSvnelobebis CamoTvla, saWiroa Sesabamisi

albaTobebis codnac. sazogadod, Tu diskretuli SemTxveviTi sididis

zrdadobiT dalagebul mniSvnelobebs aRvniSnavT x1, x2, …, xn,… simbo-

loebiT, xolo Sesabamis albaTobebs p1, p2, …, pn,… simboloebiT, maSin

vityviT, rom mocemulia diskretuli X SemTxveviTi sidide, anu moce-mulia misi ganawilebis kanoni, Tu sruldeba piroba:

pi >0, i 1 da 11

i

ip . (5.1)

diskretuli X SemTxveviTi sididis ganawilebis kanoni moxerxebulia

warmovadginoT Semdegi cxrilis saxiT:

58

cxrili 5.1.

X x1 x2 . . . xn . . .P p1 p2 . . . pn . . .

sadac rogorc aRvniSneT, pi = P{X = xi} da 11

i

ip . SevniSnoT, rom

(5.1) piroba faqtiurad imas niSnavs, rom X SemTxveviTi sidides arSeuZlia miiRos arcerTi sxva (cxrilSi miuTiTebeli) mniSvneloba.

diskretuli SemTxveviTi sididis ganawilebis kanonis codna (anu zem-

oT moyvanili cxrilis yvela komponentis codna) savsebiT sakmarisia

imisaTvis, rom gamovTvaloT am SemTxveviT sididesTan dakavSirebulinebismieri A xdomilobis albaToba. magaliTad, Tu xdomiloba mdgoma-

reobs X SemTxveviTi sididis moxvedraSi raime ricxviT (a,b] saxis in-tervalSi, anu A = {a < X b}, maSin saWiroa SevkriboT is pi albaTob-

ebi, romelTa Sesabamisi xi mniSvnelobebic moxvdeba aRniSnul interva-

lSi. sailustraciod ganvixiloT Semdegi magaliTi.

magaliTi 5.1. medikamentebis fabrikas imedi aqvs, rom mis mier

gamogonil wnevis wamals SeuZlia wnevis regulireba. maTi gaangariSeb-

iT, SemTxveviT arCeuli yoveli oTxi avadmyofisaTvis wamlis efeqtu-robis albaTobebi Semdegnairadaa ganawilebuli:

cxrili 5.2.

X 0 1 2 3 4P 0.008 0.076 0.265 0.411 0.24

aq X aRniSnavs im pacientebis raodenobas (oTxidan) romlebzec

wamalma dadebiTad imoqmeda. pirvel rigSi SevamowmoT, rom es marTl-ac albaTobaTa ganawilebis kanonia. amisaTvis SevkriboT albaTobaTa

(P) striqonSi mdgomi albaTobebi:

0.008 + 0.076 + 0.265 + 0.411 + 0.24 = 1.magaliTisaTvis, gamovTvaloT albaToba imisa, rom wamali dade-

biTad imoqmedebs or pacientze mainc. am SemTxvevaSi saZiebeli A xdo-miloba ase Caiwereba: A = {X 2}. amitom unda SevkriboT is pi albaTo-

bebi, romelTa Sesabamisi xi mniSvnelobebic metia an toli 2-is. TugavixsenebT xdomilobaTa albaTobebis Sekrebis wess wina leqciidan,gveqneba:

P{A} = P{X 2} = P{X = 2}+ P{X = 3}+ P{X = 4} ==0.265 + 0.411 + 0.24 = 0.916.

Tu SevadarebT albaTobaTa ganawilebis zemoT moyvanil cxrils

Sesabamis sixSiruli ganawilebis cxrils (romelsac Cven me-2 leqcia-

Si SevxvdiT), TvalSi gvecema aSkara msgavseba: iq fardobiT sixSireTajami toli iyo 1-isa, aqac albaTobaTa jami 1-is tolia. meores mxriv,

xdomilobis albaToba wina leqciaSi gansazRruli iyo, rogorc amave

xdomilobis fardobiTi sixSire cdaTa grZel seriaSi. xom ar metyve-

lebs es yvelaferi imaze, rom SesaZlebelia eqsperimentulad Semowmd-

59

es fabrikis mier SemoTavazebuli albaTuri modelis siswore? es mar-Tlac asea. amisaTvis saWiroa Catardes dakvirvebaTa “sakmarisad didi”

raodenoba X SemTxveviT sidideze da miRebuli fardobiTi sixSireebi

“Sedardes” Sesabamis albaTobebs. Sedarebis meTodebs Cven momavalSi

viswavliT, jer-jerobiT ki magaliTisaTvis, davuSvaT, rom 100 dakvir-vebidan (SemTxveviT amorCeul oTx-oTx avadmyofze) oTxidan zustad

erT avadmyofze wamalma dadebiTad imoqmeda 9 SemTxvevaSi, zustad oravadmyofze – 24 SemTxvevaSi, zustad sam avadmyofze – 48 SemTxvevaSi,oTxiveze – 18 SemTxvevaSi da mxolod erTxel moxda ise, rom wamal-

ma dadebiTad ver imoqmeda verc erTze SemTxveviT amorCeuli oTxi av-

admyofidan. SevadginoT Sesabamisi sixSiruli ganawilebis cxrili. masSemdegi saxe aqvs:

cxrili 5.3

avadmyofTa raodenoba 0 1 2 3 4fardobiTi sixSire 0.01 0.09 0.24 0.48 0.18

sakiTxi sixSiruli (cxrili 5.3) da albaTuri ganawilebebis(cxrili 5.2) Sedarebis Sesaxeb, anu statistikuri terminologiiT,

Tanxmobis sakiTxi, rogorc vTqviT, daskvniTi statistikis erT-erTiumniSvnelovanesi sakiTxia da mas Cven momavlisaTvis gadavdebT.

axla ki mivubrundeT diskretuli SemTxveviTi sidideebis Semd-

gom daxasiaTebas.

gansazRvra 5.3. diskretuli X SemTxveviTi sididis maTematikurlodins (mosalodnel mniSvnelobas, saSualo mniSvnelobas) uwodeben

Semdeg sidides:

11

}{i

iii

ii pxxXPxEX . (5.2)

5.1 magaliTisaTvis gvaqvs:

EX = 00.008 + 10.076 + 20.265 + 30.411 + 40.24 = 2.8,rac ise unda gavigoT, rom saSualod oTxidan 2.8 avadmyofze wamali

dadebiTad moqmedebs (ufro gasagebad, ki saSualod 40-dan 28 avadmyo-

fze wamali dadebiTad moqmedebs).

sainteresoa SevadaroT es sidide 5.3 cxrilidan gamoTvlil Se-

rCeviT saSualos:

100x 00.01 + 10.09 + 20.24 + 30.48 + 40.18 = 2.73.

SevniSnoT, rom SemTxveviTi sididis maTematikur lodins zust-

ad igive Tvisebebi aqvs, rac ricxviT monacemTa SerCeviT saSualos:

E(c1X1 c2 X2) = c1 EX1 c2 E X2 , (5.3)

nebismieri X1 da X2 SemTxveviTi sidideebisaTvis.

gansazRvra 5.4. diskretuli X SemTxveviTi sididis dispersiaewodeba Semdeg sidides:

1

2

1

22 )(}{)(i

iii

ii pxxXPxDX . (5.4)

60

jamis Tvisebebis gamoyenebiT, advili saCvenebelia, rom2

1

22 i

ii px . (5.5)

ukanaskneli formula ufro moxerxebulia gamoTvlebisaTvis.

gansazRvra 5.5. diskretuli X SemTxveviTi sididis standartu-li gadaxra ewodeba ariTmetikul kvadratul fesvs misi dispersiidan:

2

1

2

1

2)( i

iii

ii pxpxDX . (5.6)

SevniSnoT, rom SemTxveviTi sididis dispersiasac zustad igive

Tvisebebi aqvs, rac ricxviT monacemTa SerCeviT dispersias:

1) Dc = 0; 2) D(c1X1 c2 X2) = 2221

21 DXcDXc ,

damoukidebeli X1 da X2 SemTxveviTi sidideebisaTvis, anu iseTi SemTx-

veviTi sidideebisaTvis, romelTaTvisac

P{X1A, X2B}= P{X1A} P{X2B}, (5.7)nebismieri A da B ricxviTi simravleebisaTvis.

5.1 magaliTisaTvis gvaqvs:

2 = 00.008 + 10.076 + 40.265 + 90.411 + 160.24 - 2.82 = 1.835,

saidanac 36.1835.1 .

SemTxveviTi sidideebis es ricxviTi maxasiaTeblebi moxerxebu-

lia imisaTvis, rom warmodgena Segviqmnan SemTxveviTi sidididis cval-

ebadobis Sesaxeb. magaliTad, bevri (magram ara yvela) SemTxveviTi si-

dide 0.95-ze meti an toli albaTobiT meryeobs intervalSi

[ - 2; + 2]. (5.8)

marTlac, Cveni magaliTis SemTxvevaSi, = 2.8, = 1.36. amitom[ - 2; + 2] = [2.8 - 21.36; 2.8 + 21.36] = [0.08;5.52]. 5.2 cxrilismixedviT, am intervalSia moTavsebuli X SemTxveviTi sididis

mniSvnelobebi 1, 2, 3 da 4 da maSasadame,

P{0.08 X 5.52}= P{X = 1}+ P{X = 2}+ P{X = 3}+ P{X = 4}=0.076+0.265+0.411+0.24 = 0.992 0.95.

5.2. diskretuli SemTxveviTi sididis ganawilebis funqcia.

gansazRvra 5.6. X SemTxveviTi sididis ganawilebis funqcia(aRiniSneba FX(x) simboloTi) ganimarteba rogorc {Xx} xdomilobisalbaToba nebismieri namdvili x-Tvis, anu

FX(x) = P{X x}. (5.9)

rogorc {X x} tipis xdomilobis albaTobas, FX(x) ganawilebisfunqcias aqvs Semdegi Tvisebebi:

1) 0 FX(x) 1, nebismieri x-saTvis, x R;2) FX(-) = 0, FX(+) =1;3) FX(x) aris x cvladis araklebadi funqcia;

61

4) FX(x) ganawilebis funqcia uwyvetia marjvnidan.5.1 magaliTis mixedviT gavarkvioT, Tu rogori grafiki aqvs di-

skretuli SemTxveviTi ganawilebis funqcias. cxadia, rom sanam x<0 ga-

nawilebis funqciis mniSvneloba 0-is tolia; roca 0 x<1, P{X x}=0.008; roca 1 x<2, albaTobaTa Sekrebis wesis gamo, P{X x}= 0.008 +0.076 =0.084; analogiurad, roca 2 x<3, P{X x}= 0.008 + 0.076 + 0.265= 0.349; roca 3 x<4, P{X x}= 0.008 + 0.076 + 0.265 +0.411 = 0.76 dabo-

los, roca x 4, P{X x}= 0.008 + 0.076 + 0.265 +0.411 +0.24 = 1, anumonacemebis empiriuli ganawilebis funqciis msgavsad, FX(x) funqcias

aqvs uban-uban mudmivi (safexura) funqciis grafiki:

SevniSnoT, rom sazogadod diskretuli SemTxveviTi sididis ga-

nawilebis funqciisaTvis

FX(xi) = P{X xi}= P{X = x1} + P{X = x2} +…+ P{X = xi} = = p1 + p2 +…+ pi (5.10)

da amitom

pi = FX(xi) - FX(xi -1), yoveli i-Tvis, i 1 (aq x0 = 0). (5.11)(5.10) da (5.11) Tanafardobebi gveubneba, rom diskretuli SemT-

xveviTi sididis ganawilebis kanonisa da misi ganawilebis funqciis

mocema eqvivalenturia.

5.3. gadanacvlebebi da jufdebebi.

vTqvaT, n da k raime naturaluri ricxvebia da k n. davsvaTaseTi SekiTxvebi: a) ramdennairad SeiZleba n elementiani simravlidan

k elementiani qvemimdevrobis amorCeva? b) ramdennairad SeiZleba nelementiani simravlidan k elementiani qvesimravlis amorCeva?

SevniSnoT, rom SekiTxvebs Soris aris gansxvaveba: pirvel SemT-

xvevaSi laparakia qvemimdevrobebis amorCevaze, maSin roca, meore SemT-

62

xvevaSi saubaria qvesimravleebis amorCevaze. igulisxmeba, rom magali-Tad, mimdevrobebi (1, 2, 3) da (3, 1, 2) gansxvavebuli mimdevrobebia, mag-

ram rogorc simravleebi {1, 2, 3} da {3, 1, 2} erTidaigive simravleebia.

amitom gasagebia, rom pasuxebi zemoT dasmul kiTxvebze iqneba sxvadas-

xva. pirvel SemTxvevaSi TiToeul qvemimdevrobas n elementidan k ele-

mentian gadanacvlebas uwodeben, xolo meoreSi – TiToeul qvesimrav-

les n elementidan k elementian jufdebas uwodeben. SevecadoT axla

gamoviangariSoT am SesaZleblobaTa (gadanacvlebaTa da jufdebaTa) ra-

odenobebi.

daviwyoT gadanacvlebebiT. n elementiani simravlidan qvemimdev-

robis pirveli elementis arCeva SesaZlebelia zustad n gansxvavebuligziT (SesaZlebelia arCeul iqnas mocemuli simravlis yoveli elemen-

ti). meore nabijze ukve amorCeva xdeba darCenili (n-1)-elementiani

simravlidan; ase, rom amis gakeTeba SeiZleba mxolod n -1 gansxvavebu-

li gziT. amrigad, qvemimdevrobis pirveli ori wevris arCeva SesaZleb-

elia n(n-1) gansxvavebuli gziT. Tu gavagrZelebT msjelobas k-ur nab-ijamde, advilad davrwmundebiT, rom n elementiani simravlidan k el-

ementiani qvemimdevrobis amorCeva SeiZleba

( 1) ( 1)knP n n n k (5.12)

gansxvavebuli gziT.

SevniSnoT, rom (5.12) formulis mixedviT,

( 1) 2 1 !nnP n n n (5.13)

aris n elementiani simravlis elementTa yvela SesaZlo gadanacvleba-Ta raodenoba (simbolo n! ikiTxeba: n faqtoriali da is warmoadgens 1-

dan n-mde (CaTvliT) yvela naturaluri ricxvis namravls. miRebulia,

rom 0! = 1). SevniSnoT, romn! = n (n -1)! = n (n -1)(n -2)! = = n (n -1) (n – k +1)(n -k)!. (5.14)

(5.12) da (5.14)-dan advilad davaskvniT, rom

!/( )!knP n n k . (5.15)

gadavideT axla jufdebaTa raodenobis gamoTvlaze. cxadia, rom

n elementiani simravlidan k elementian gadanacvlebaTa raodenoba im-

denjer metia Sesabamis jufdebaTa raodenobaze, ramdennairadac SeiZl-

eba k elementiani simravlis yvela elementis yvelanairi gadanacvleba.

es ki, rogorc vnaxeT, SesaZlebelia k! sxvadasxvanairad. amitom n el-

ementiani simravlidan k elementian jufdebaTa raodenoba (romelsacknC -Ti aRvniSnavT) gamoiTvleba Semdegi formuliT:

)!(!

!

! knk

n

k

PC

knk

n . (5.16)

cxadia, rom knC = kn

nC , yoveli k-saTvis, 0 k n.

63

Semotanil sidideebs erTobliobaSi kombinatorikis elementebsuwodeben. maTi gamoangariSebis sailustraciod mogvyavs ramdenime mar-

tivi savarjiSo:

4! = 1234 = 24; 5! = 5 4! = 5 24 = 120;

336678!5

!5678

!5

!8

)!38(

!838

P ;

3557321

567

!3!4

!7

)!47(!4

!747

C .

5.4. binomuri ganawileba.

albaTobaTa binomuri ganawileba ganawileba aqvs nebismier disk-

retul SemTxveviT sidides, romelic n damoukidebeli cdis Catarebis

dros iTvlis raime A xdomilobis moxdenaTa (warmateba-Ta) raodenob-

as da romlis moxdenis albaToba yovel cdaSi erTnairia da aris p == P(A). aRvniSnoT aseTi SemTxveviTi sidide Sn(A)-Ti. maSin misi SesaZ-lo mniSvnelobebia 0,1,2,…,n, xolo Sesabamisi albaTobebi iTvlebaformuliT:

{ ( ) } ( ) ( ) (1 ) , 0,...,n kkk C k k n k

n n nP S A k C P A P A C p p k n . (5.17)

am ganawilebis gamoyenebis sailustraciod gavarCioT Semdegi

magaliTi 5.2. cnobilia, rom biWis dabadebis albaTobaa p = 0.51.ras udris albaToba imisa, rom 7 axalSobilidan iqneba 5 biWi, TuSobadobebi damoukidebelia?

amoxsna. amocanis pirobis Tanaxmad, n = 7, k = 5 da p = P(A) = 0.51(cxadia, aq A aRniSnavs xdomilobas imisa, rom axalSobili iqneba bi-

Wi). amitom (5.17) formulis Tanaxmad, gveqneba:

391.049.051.02149.051.0}5)({ 2525577 CASP .

sainteresoa, Tu rogoria am ganawilebis (an rac igivea Sn(A) Se-mTxveviTi sididis) ricxviTi maxasiaTeblebi: maTematikuri lodini da

dispersia (standartuli gadaxra).

binomuri SemTxveviTi sididis maTematikuri lodini tolia cda-

Ta ricxvis namravlisa warmatebis P(A) albaTobaze:

E Sn(A) = n P(A) = np. (5.18)

binomuri SemTxveviTi sididis dispersia gamoiTvleba SemdegiformuliT:

D Sn(A) = n P(A) (1- P(A)) = np(1- p). (5.19)

maSin gasagebia, rom Sesabamisi standartuli gadaxra tolia:

)1( pnp . (5.20)

am ganawilebis modas xSirad ualbaTes ricxvs uwodeben. is wa-rmoadgens k-s im mniSvnelobas, romlisaTvisac (5.17) albaToba udides

64

mniSvnelobas Rebulobs. mas m0-iT aRniSnaven da misTvis samarTlianiaSemdegi utoloba:

(n + 1) p – 1 m0 (n + 1) p. (5.21)

SevniSnoT, rom radgan [(n + 1) p – 1 , (n + 1) p] ricxviTi inter-valis sigrZe 1-is tolia, arsebobs erTi mainc naturaluri m0, rome-

lic mas ekuTvnis, anu ualbaTesi ricxvi yovelTvis arsebobs. garda

amisa, Tu (n + 1) p naturaluri ricxvia, maSin ualbaTesi ricxvi

oria, m0 = (n + 1) p an m0 = (n + 1) p – 1 anu am SemTxvevaSi binomuri

ganawileba bimodaluria.zogjer saWiroa (5.17) formulis magivrad gamoviyenoT misi gam-

oTvlis mimdevrobiTi, e.w. rekursiuli (rekurentuli) gadaTvlis we-si, romelic SemdegSi mdgomareobs:

}{)1)(1(

)(}1{ kSP

pk

pknkSP nn

. (5.22)

magaliTi 5.3. 1986 wels 60-dan 64 wlamde 100 mamakaci acriliqna gripis sawinaaRmdego axali vaqciniT. erTi wlis Semdeg xuTi ma-

Tgani gardaicvala. niSnavs Tu ara es, rom axali vaqcina uvargisia,

Tu am asakSi myofi xuTi adamianis erT weliwadSi gardacvalebaSi

“uCveulo” araferia?

amis dasadgenad, sikvdilianobis cxrilebidan moZiebul iqna 60-

64 ww. asakobrivi intervalisaTvis mamakacis erT weliwadSi gardacva-

lebis albaToba, rac p = 0.02-is toli aRmoCnda. maSin (5.17)-is gamoye-

nebiT Cven SegviZlia gamovTvaloT albaToba imisa, rom gardacvlilTaraodenoba iqneba k-s toli, nebismieri k –saTvis, k = 0,1, …,100. magramis, rac Cven gvainteresebs aris, Tu ramdenad mosalodneli iyo xuTi

an meti adamianis gardacvaleba, anu {Sn 5} xdomilobis albaToba, rac

albaTobaTa Sekrebis wesis Tanaxmad, tolia

P{Sn 5}=

100

5

100100

100

5

)98.0()02.0(}{k

kkk

kn CkSP .

cxadia, rom gamoTvlebis gamartivebis mizniT, umjobesia {Sn 5} xdom-

ilobis damatebiTi {Sn < 5} xdomilobis albaTobis gamoTvla da Semdeg

damatebiTi xdomilobis albaTobis formulis gamoyeneba:

P{Sn 5}= 1 - P{Sn < 5}==1- P{Sn =0}- P{Sn =1}- P{Sn =2}- P{Sn =3}- P{Sn =4}.

(5.17) formulis Tanaxmad, P{Sn = 0} = (0.98)100 0.13262. analog-

iurad, (5.22) formulis gamoyenebiT miviRebT, rom:

27065.013262.098.0

02.0100}0{

98.0)10(

02.0)0100(}1{ 100100

SPSP ;

27341.027065.098.02

02.099}1{

98.0)11(

02.0)1100(}2{ 100100

SPSP ;

65

18228.027341.098.03

02.098}2{

98.0)12(

02.0)2100(}3{ 100100

SPSP ;

09021.018228.098.04

02.097}3{

98.0)13(

02.0)3100(}4{ 100100

SPSP .

amitom sabolood gvaqvs:

P{Sn 5}= 1- P{Sn =0}- P{Sn =1}- P{Sn =2}- P{Sn =3}- P{Sn =4}== 1 – 0.13262 – 0.27065 – 0.27341 – 0.18228 – 0.09021 = 0.051.radgan 0.051 arc ise “mcire” albaTobaa, davaskvniT, rom xuTi

kacis gardacvalebiT arc ise “uCveulo” ram momxdara. Tu magaliTad,

gardaicvleboda 10 da ara 5 kaci, maSin analogiuri gamoTvlebiT gve-

qneboda, rom P{Sn 10}< 0.001, rac faqtiurad ar unda momxdariyo daamitom aseTi vaqcina saerTod unda amogveRo xmarebidan.

5.5. puasonis ganawileba.

es ganawileba, rogorc wesi, Cndeba iSviaTi xdomilobebis Sesw-

avlis dros. ganvixiloT magaliTad, muclis tifiT gardacvalebaTa

raodenoba sakmarisad didi droiTi intervalis, magaliTad, erTi wlis

ganmavlobaSi. Tu davuSvebT, rom am avadmyofobiT yoveldRiuri garda-

cvalebis albaToba sakmarisad mcirea da TanaukveT intervalebSi gard-

acvalebebi damoukidebeli SemTxveviTi sidideebia, maSin erTi wlis ga-

nmavlobaSi gardacvalebaTa raodenobas eqneba puasonis ganawileba. saz-ogadod, diskretul X SemTxveviT sidides aqvs puasonis ganawileba,Tu misi SesaZlo mniSvnelobebia 0,1,2,…, xolo Sesabamisi albaTobebiiTvleba Semdegnairad:

ek

kXPk

!}{ . (5.23)

gavarkvioT rogoria gardacvalebaTa raodenobis ganawileba dak-

virvebis dawyebidan drois t momentamde (t sakmarisad didi ricxvia)?ganvixiloT [0, t] intervalis mcire droiTi qveintervali t. davuSvaT,rom Sesrulebulia Semdegi pirobebi:

I. zustad erTi gardacvalebis albaToba am qveintervalSi prop-

orciulia intervalis sigrZis:

P{t intervalSi moxdeba erTi gardacvaleba} t,sadac raime dadebiTi mudmivia;

II. P{t intervalSi ar moxdeba arcerTi gardacvaleba}1 - t.faqtiurad, es ori piroba niSnavs, rom t intervalSi ar xdeba erT

gardacvalebaze meti.

III. gardacvalebaTa es procesi stacionarulia droSi (mudmivia,

sadac ar unda aviRoT t sigrZis mqone qveintervali);

66

IV. drois TanaukveT intervalebSi gardacvalebaTa raodenobebidamoukidebeli SemTxveviTi sidideebia (anu gardacvalebaTa raodenoba

romelime intervalSi gavlenas ar axdens gardacvalebaTa albaTobebze

sxva TanaukveT intervalebSi).

maSin [0, t] intervalSi gardacvalebaTa raodenobas aqvs puasonis

ganawileba, romlisaTvisac (5.23) formulaSi Semavali parametri

akmayofilebs pirobas:

= t. (5.24)magaliTi 5.4. davuSvaT, rom muclis tifiT gardacvalebaTa ra-

odenoba erTi wlis ganmavlobaSi ganawilebulia puasonis kanoniT, pa-

rametriT = 4.6. rogoria gardacvalebaTa albaTuri ganawileba eqvsi

Tvis ganmavlobaSi? sami Tvis ganmavlobaSi?

amoxsna. X-iTa da Y-iT aRvniSnoT garacvalebaTa raodenoba Ses-

abamisad, eqvssa da sam TveSi. radgan = 4.6 da t = 1 (5.24) formulid-

an gamomdinareobs, rom = 4.6. eqvsi Tvis ganmavlobaSi = 4.6, t = 0.5 = t = 4.6 0.5 = 2.3, xolo sami Tvis ganmavlobaSi = 4.6, t = 0.25 = t = 4.6 0.25 = 1.15. ase, rom X SemTxveviT sidides aqvs (5.23)ganawileba parametriT = 2.3, xolo Y SemTxveviT sidides aqvs (5.23)ganawileba parametriT = 1.15.

sainteresoa rogoria sazogadod puasonis ganawilebis ricxviTi

maxasiaTeblebi: maTematikuri lodini da dispersia.

puasonis ganawilebis maTematikuri lodini da dispersia toliada emTxveva ganawilebis parametrs:

EX = DX = . (5.25)

SevniSnoT, rom binomuris ganawilebis msgavsad, aqac SesaZleb-elia albaTobaTa gadaTvlis rekursiuli wesis dadgena. marTlac,

(5.23)-dan advilad miviRebT, rom

}{1

}1{ kXPk

kXP

. (5.26)

mivubrundeT 5.4 magaliTs da gamovTvaloT magaliTad, albaTob-

ebi P{Y = 1}, P{Y = 2}, P{Y = 3}, P{Y = 0}, Tu cnobilia, rom e-1.15 0.3166. gvaqvs: P{Y = 0}= e-1.15 0.3166;

3641.03166.015.1}0{10

15.1}1{

YPYP ;

2094.03641.02

15.1}1{

11

15.1}2{

YPYP ;

0803.02094.03

15.1}2{

12

15.1}3{

YPYP .

puasonis albaTobebis grafikuli warmodgenisaTvis SevniSnoT,

rom (5.26) formulis Tanaxmad, yoveli Semdegi albaToba winaze pata-

67

raa dawyebuli k = 1-dan, magram imisdamixedviT, Tu rogoria ganawile-

bis parametri (1-ze didia, Tu patara), SesaZlebelia ori SemTxveva:

P{X = 0}= e- e- = P{X = 1}, roca 1 da piriqiT, P{X = 0}> P{X =1}, roca < 1. amitom grafikebs Semdegi saxe aqvs:

5.6. kavSiri binomur da puasonis ganawilebebs Soris.

rodesac cdaTa ricxvi binomuri ganawilebisaTvis “sakmarisad

didia”, xolo warmatebis albaToba – “sakmarisad mcire”, binomuri ga-

nawileba SegviZlia SevcvaloT puasonis ganawilebiT, romlis paramet-

ric tolia = np, Tu es ukanaskneli arc “didia” da arc “Zalianmcire”, sxva sityvebiT, rom vTqvaT,

nkek

npppC np

kknkk

n ,...,2,1,0,!

)()1( . (5.27)

sailustraciod moviyvanoT Semdegi

magaliTi 5.5. davuSvaT Cven gvainteresebs gadadis Tu ara gene-

tikurad mkerdis kibo qalebSi. vTqvaT, 40-49ww. asakobriv jgufSi

myofi 1000 qalidan, romelTa dedebsac hqondaT adre es daavadeba,

oTxs ganuviTarda daavadeba. ramdenad uCveuloa es, Tuki wina gamokv-

levebidan cnobilia, rom 1000-dan saSualod erTs uviTardeboda aRni-

Snuli daavadeba?

amoxsna. Cven gvainteresebs {X 4} xdomilobis albaToba (sadac

X-iT aRniSnulia memkvidreebSi daavadebaTa raodenoba), roca n =1000, p= 0.001. saZiebeli albaTobaa P{X 4} = 1 - P{X 3} = 1 - P{X = 0}- P{X =1}- P{X = 2}- P{X = 3}. magram binomuri albaTobebis gamosaxulebebis

gamoangariSeba sakmarisad rTulia da amitom gamovTvaloT zemoT moyv-

anili albaTobebi puasonis formulis gamoyenebiT, romlisaTvisac == np= 10000.001=1. miviRebT,

68

P{X = 0} 0.3679, P{X = 1} 0.3679, P{X = 2} 0.1839, P{X = 3} 0.0613.amitom sabolood gveqneba

P{X 4} = 1 - P{X = 0}- P{X = 1}- P{X = 2}- P{X = 3}== 1 – (0.3679+0.3679+0.1839+0.0613) = 0.019.

am albaTobis simcire asabuTebs eWvs am daavadebis memkvidreob-

iT xasiaTze.

amocanebi1. ramdennairad SeiZleba anginis 50 SemTxvevidan 5-is amorCeva,

Tu dalagebas yuradReba eqceva?

2. gaakeTeT amocana 1 im SemTxvevisaTvis, roca dalagebas yura-

dReba ar eqceva.

3. gamoTvaleT 1010

910

810

710

610

510

410

310

210

110

010 ,,,,,,,,,, PPPPPPPPPPP .

4. gamoiangariSeT 1010

910

810

710

610

510

410

310

210

110

010 ,,,,,,,,,, CCCCCCCCCCC .

5. daamtkiceT, rom 11

1

kn

kn

kn CCC nebismieri 0 k n –1.

6. gamoTvaleT albaToba imisa, rom 5.3 magaliTSi gardacvlil-

Ta raodenoba meti an tolia 7-ze, 8-ze, 9-ze.

7. ras udris albaToba arcerTi adamianis gardacvalebisa 5.3

magaliTSi?

8. gamoTvaleT Semdegi saxis albaTobebi 5.4 magaliTSi:P{X = 0}, P{X = 1}, P{X = 2}, P{X = 3}, P{X = 4}, P{X = 5}.

9. gamoTvaleT me-8 amocanis albaTobebi rekursiulad.

10. cnobilia, rom Zroxebis 5% daavadebulia tuberkuloziT.

SemTxveviT SearCies 10 Zroxa sxvadsxva regionidan da daTvales daava-

debuli Zroxebis raodenoba. dawereT daavadebuli Zroxebis ganawileb-

is kanoni.

11. depresieuli pacientebis 60% klasificirebuli iyo rogorc

introvertuli pirovneba, 40% ki rogorc eqstrovertuli. dawereT

SemTxvevioT SerCeul 15 pacientSi introvertuli pirovnebebis ganawi-

lebis kanoni.

12. 10 ZaRlis sisxlSi daTvales wiTeli ujredebis raodenoba(1000000 erT ml-ze) da miiRes Semdegi monacemebi:

9.5 7.5 7.0 5.9 9.5 6.2 6.5 8.7 5.8 7.7.

12.1. ZaRlebis ramden procents aqvs sisxlSi wiTeli ujredebis rao-

denoba naklebi an toli 6.0-ze;

12.2. Tu ZaRlebis es SerCeva aris ZaRlebis reprezentatuli SerCeva,

maSin ras udris albaToba imisa, rom SemTxveviT SerCeuli 20 ZaRli-

dan 4-s aqvs wiTeli ujredebis raodenoba naklebi an toli 6.0-ze.

13. afrikaSi 45 ubanze ikvlevdnen lomebis arsebobas (“+” niSn-

avs, rom lomebi arian, xolo “-” ki lomebi ar arian)":1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

69

- - - + - + + - - - + - - + -16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

+ - - - - - - + - - - - - - -31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

- - - + + - - - - - - - - - -

13.1. ubnebis ramden procentze iyo lomebis arseboba dafiqsirebuli.

13.2. ras udris albaToba imisa, rom SemTxveviT SerCeuli 12 ubnidan6-ze aRmoCndeba lomi.

14. cnobilia, rom arcerTi soko ar cocxlobs momaval wlamde.

nebismieri soko momdevno wels iZleva raodenobis axal sokos. da-

vuSvaT, rom mimdinare wels xarobs ori soko. vipovoT momaval wels

sokoTa raodenobis ganawilebis kanoni, gamovTvaloT -s maTematik-

uri lodini da dispersia, Tu SemTxveviTi sididis ganawilebis kano-

nia:

0 1 2

P 0.2 0.6 0.2

m i T i T e b a: SeadgineT damoukidebeli , wyvilis erToblivi

ganawilebis kanoni, .

l e q c i a 5.

Tavi 6. uwyveti SemTxveviTi sidideebi. ZiriTadi

albaTuri ganawilebebi.

6.1. uwyveti SemTxveviTi sidideebi da maTi ganawileba.

wina leqciaSi Cven gavecaniT diskretul SemTxveviTi sidideebs

da aRvniSneT, rom maTi mocema niSnavda an maTi ganawilebis an Sesabam-isi ganawilebis funqciis mocemas, romelsac, sxvaTa Soris, hqonda sa-

fexura funqciis grafiki. samarTliania piriqiTac, Tu raime SemTxvevi-

Ti sididis ganawilebis funqcia uban-uban mudmivia, maSin Sesabamisi

SemTxveviTi sidide diskretulia. magram arsebobs SemTxveviTi sidedee-

bi, romelTa ganawilebis funqcia aRaraa uban-uban mudmivi. magaliTisa-

Tvis, SevecadoT davaxasiaToT iseTi sidide, rogoricaa SemTxveviT ar-

Ceuli adamianis asaki mocemul momentSi.

magaliTi 6.1. davuSvaT zRvruli asaki Seadgens weliwads (ma-

galiTad, = 100 an = 150, Tumca bibliuri personaJebi ufro di-

dxansac cocxlobdnen). SemoviRoT SemTxveviTi sidide T – SemTxveviT

arCeuli adamianis asaki 2009 wlis 1 ianvrisaTvis da vigulisxmoT,

rom asaki izomeba zustad (wlebSi, TveebSi, dReebSi, saaTebSi, wuTeb-

70

Si, wamebSi da a.S.). maSin cxadia, rom am SemTxveviT sidides SeuZlia

miiRos yvela namdvili mniSvneloba [0,] ricxviTi intervalidan. mag-

ram gansxvavebiT diskretuli SemTxveviTi sidideebisagan, aq azri ara

aqvs vilaparakoT {T = x} xdomilobasa da mis albaTobaze, radgan wina-

swar veravin ityvis ama Tu im adamianis asaks (zemoT aRniSnuli skal-

is mixedviT), aramed, ufro azriani iqneba vilaparakoT {a T b} an

{T x} saxis xdomilobebsa da maT albaTobebze. am SemTxvevaSic T Sem-TxveviTi sididis ganawilebis funqcias uwodeben {T x} saxis xdomi-

lobis albaTobas:

FT(x) = P{T x}. (6.1)albaT yvela gaiziarebs im mosazrebas, rom nebismieri asakis

adamians Tanabari Sansi aqvs iyos arCeuli da Sesabamisad, ganawileb-

is funqciis mniSvneloba 0-is tolia, roca x<0, is tolia FT(x) == P{T x}= x/, roca 0 x da tolia 1-s, roca x > . ase, rom Se-sabamisi ganawilebis funqciis grafiks aqvs Semdegi saxe:

FT

1

0 x

am ganawilebas uwodeben Tanabar ganawilebas [0,] Sualedze.aseve ganimarteba Tanabari ganawileba nebismier [c,d] Sualedze:

dx

dxccd

cxcx

xFX

roca

roca

roca

,1

,

,0

)( . (6.2)

SevniSnoT, rom Tanabari ganawilebis funqcia ar warmoadgens

safexura funqcias da is uwyvetia. aseT SemTxvevebSi amboben, rom T(X) SemTxveviT sidides aqvs uwyveti ganawileba. am dros {T = x} xdom-

ilobis albaToba 0-is tolia nebismieri x-saTvis, magram intuiciuradmainc gvesmis, rom SemTxveviTi sididis esa Tu is mniSvneloba ufro

metadaa mosalodneli, vidre romelime sxva mniSvneloba. SevecadoTsicxade SevitanoT am sakiTxSi. amisaTvis SemovitanoT ganawilebis si-mkvrivis cneba, romelic ganmartebiT iseTi arauaryofiTi funqciaa,

romlis qveS raime [a,b] ricxviTi intervalis farTobic emTxveva SemT-

xveviTi sididis aRniSnul intervalSi moxvedris albaTobas. analizu-

rad, Tu cnobilia SemTxveviTi sididis ganawilebis funqciis saxe, Se-

71

sabamisi simkvrive (romelic fX(x) simboloTi aRiniSneba) warmoadgensganawilebis funqciis warmoebuls:

fX(x) = )(xFX . (6.3)

sailustraciod moviyvanoT simkvrivis Semdegi

magaliTi 6.2. maRalwneviani avadmyofebisaTvis:

fX

0 80 90 95 100 x

naxazze gamosaxuli A, B da C areebis farTobebi Sesabamisad Se-

esabameba sazRvris (borderline), Sua hipertonul (mild hypertensive) daZlier hipertonulobis (severe hypertensive) albaTobebs 35-44 ww. asa-

kobriv jgufSi myofi mamakacebisaTvis.

am farTobebis (albaTobebis) ricxviTi mniSvnelobebis gamoTvla,

dakavSirebulia imasTan, Tu rogor wirs warmoadgens ganawilebis simk-

vrive, anu rogoria misi analizuri saxe, gamoTvla ki xdeba maTematik-

ur analizSi kargad cnobili niuton-laibnicis integraluri formu-liT:

)()()(}{ aFbFdxxfbXaP XX

b

a

X . (6.4)

diskretuli SemTxvevis msgavsad, aqac Semodis uwyveti SemTxve-

viTi sididis ricxviTi maxasiaTeblebi, maTematikuri lodini da dispe-

rsia (standartuli gadaxra). imisaTvis, rom ar gadavitvirToT maTema-

tikuri simboloebiT, aRvniSnavT, rom maTi Sinaarsi igivea, rac diskr-

etul SemTxvevaSi, mxolod gamoangariSebis wesisaTvis davimaxsovroT,

rom diskretulisaTvis moyvanil yvela formulaSi, ajamvis simbo-

lo icvleba integrebis simboloTi, SemTxveviTi sididis xi mniSvne-

loba icvleba mimdinare cvladis x mniSvnelobiT da bolos pi albaTo-

ba icvleba fX(x)dx gamosaxulebiT. magaliTad, maTematikuri lodinisgamosaxulebebi diskretul da uwyvet SemTxvevebSi Sesabamisad aris:

diskretuli SemTxveva uwyveti SemTxveva

1i

ii pxEX

dxxfxEX X )(

AB C

72

(6.3)-dan gamomdinare, SevniSnoT, rom fX(x)dx marTlac albaTobaaX SemTxveviTi sididis moxvedrisa mcire [x, x + dx] intervalSi:

fX(x)dx P{x < X x + dx}. (6.5)

magaliTi 6.1 (gagrZeleba). Tanabari ganawilebis zemoT SemoRe-

buli funqciisaTvis, ganawilebis simkvrives aqvs Semdegi saxe:fT

1/

0 a b x

T SemTxveviTi sididis [a,b] ricxviTi intervalSi moxvedris

albaToba, rogorc zemoT aRvniSneT, tolia naxazze gamosaxuli A aris

farTobisa, romelic am SemTxvevaSi warmoadgens marTkuTxeds, romlis

gverdebis sigrZeebia 1/ da b - a. ase, romP{a < T b} = S(A) = (1/ )( b - a) = ( b - a) /.

SevniSnoT, rom albaToba ar Seicvleba, Tu [a,b] intervals “ga-

vwevT” marcxniv (cxadia, 0-mde) an marjvniv (-mde). sxva sityvebiTrom vTqvaT, Tanabari sigrZis intervalebSi moxvedris albaTobebi Ta-

nabaria. swored amitomac hqvia am ganawilebas Tanabari ganawileba. Tu

magaliTad, =100, a = 67 da b = 82, maSin albaToba imisa, rom SemTxve-

viT amorCeuli adamianis asaki moxvdeba [67,82] asakobriv intervalSi

tolia: P{67 < T 82} = (82 - 67)/100 = 15/100 = 0.15.

6.2. normaluri SemTxveviTi sidideebi (normaluri ganawileba).

es aris yvelaze xSirad gamoyenebadi uwyveti ganawileba. masxSirad gausis ganawilebasac uwodeben (germaneli maTematikosis gaus-

is sapativsacemod). am kanons emorCileba uamravi SemTxveviTi sidide,

maT Soris axalSobilTa wonebi 2.1 magaliTidan, sisxlis wneva 2.3 ma-

galiTidan da uamravi sxva. amitomac momavalSi xSirad iqneba daSvebu-

li Sesaswavli statistikuri modelis normaluroba.

gansazRvra 6.1. normaluri ewodeba iseT uwyvet X SemTxveviT

sidides, romelic Rebulobs namdvil mniSvnelobebs da romlis ganawi-

lebis simkvrives aqvs Semdegi saxe:

2

2

2

)(exp

2

1)(

x

xf X . (6.6)

A

73

(6.6) ganawilebis simkvrives is Tviseba aqvs, rom fX( + t) = fX( - t),rac mis simetrulobas niSnavs parametris mimarT. SevniSnoT, rom

fX(+) = fX(-) = 0 da udides mniSvnelobas fX(x) funqcia aRwevs x =

wertilSi:

2

1)( Xf , (anu am ganawilebis modaa x = ). sabolo-

od, mis grafiks aqvs zariseburi forma da gamosaxulia naxazze:

fX

21

0 - +

rogorc vxedavT, normaluri ganawilebis simkvrivis saxes sav-

sebiT gansazRvravs misi parametrebi: da > 0: gviCvenebs ganawil-

ebis adgilmdebareobas ricxviT RerZze, xolo parametri gviCvenebs,

Tu ramdenad maRali (da maSasadame mWidroa grafiki -s mimarT), anpiriqiT, Tu ramdenad dabali da gawolilia is Ox RerZis gaswvriv.es faqtebi sulac araa SemTxveviTi, radgan am parametrebis Sinaarsi

SemdegSi mdgomareobs:

EX = da DX = 2. (6.7)

maSasadame, yofila X SemTxveviTi sididis maTematikuri lodini,xolo warmoadgens X SemTxveviTi sididis standartul gadaxras. imfaqts, rom X SemTxveviTi sidide normaluradaa ganawilebuli parame-

trebiT da 2 mokled ase weren:

X ~ N( , 2). (6.8)

6.2.1. normaluri ganawilebis Tvisebebi.

1) normaluri X SemTxveviTi sididis (N(,2)-is) nebismieri

wrfivi gardaqmna Y = aX + b kvlav normaluria da

Y ~ N(a + b, a22). (6.9)

2) Tu 1) TvisebaSi aviRebT a = 1/ da b = - /, maSin 1)-danmiviRebT, rom

Y = (X - ) / ~ N(0, 1). (6.10)unda aRiniSnos, rom X SemTxveviTi sididis aseT wrfiv gardaqm-

nas mis standartizacias uwodeben, xolo miRebul N(0,1) ganawilebas

ki standartul normalur ganawilebas. am ganawilebas fundamentalu-

ri adgili ukavia mTels daskvniT statistikaSi. misi saSualebiT xde-

74

ba normalur ganawilebasTan dakavSirebuli yvela albaTobis ricxviTimniSvnelobis gamoTvla. es gamowveulia e.w. standartuli normaluricxrilebis arsebobiT: daskvniTi (maTematikuri) statistikis nebismiersaxelmZRvaneloSi moyvanilia standartuli normaluri ganawilebis

cxrilebi, anu (x) P{N(0,1) x} saxis albaTobebi x cvladis nebism-

ieri arauaryofiTi mniSvnelobisaTvis, romlisaTvisac 0 x 3 (zogjerx 4-mde). mxolod arauaryofiT x-ebSi ganawilebis funqciis mniSvnel-

obebis mocema ganpirobebulia imiT, rom uaryofiT x-ebSi albaTobebi

iTvleba (x) funqciis Semdeg Tvisebaze dayrdnobiT, rom(-x) = 1 - (x). (6.11)

garda amisa, (3) 0.9987, rac savsebiT amarTlebs funqciis

mniSvnelobebis mocemas mxolod 0 x 3 intervalisaTvis, rac sakmar-

isia gamoTvlebis Casatareblad, Tumca ufro zusti gamoTvlebisaTvis

cxrilebi rogorc vTqviT, Sedgenilia 4-mdec ki; magaliTad, (3.62) 0.9999. standartuli normaluri ganawilebis simkvrives aRniSnaven

(x)-iT (nacvlad zemoT naxmari fX(x)-isa).aRvniSnoT , (x) P{N(,2 ) x}. maSin 2) Tviseba SeiZleba

ase gadavweroT:

, (x) = ((x - )/). (6.12)

2) Tvisebis (an rac igivea (6.12) formulis) gamoyenebis sailu-

straciod gavagrZeloT 6.2 magaliTis ganxilva da davsvaT aseTi Sek-

iTxva: ras udris albaToba imisa, rom Sua hipertonul (mild hypertens-ive) 35-44 ww. asakobriv jgufSi myofi mamakacebis diastoluri sisx-

lis wneva moxvdeba intervalSi [95,100]?amoxsna. aRniSnuli asakis Sua hipertonuli (mild hypertensive)

avadmyofebisaTvis cnobilia, rom = 90 da =15. aRvniSnoT X-iT aRn-iSnuli asakisaTvis diastoluri sisxlis wnevis sidide, CavTvaloT,

rom is ganawilebulia normalurad zemoT aRniSnuli parametrebiT,

anu X ~ N(90, 225). maSin standartuli normaluri ganawilebis cxril-

ebis gamoyenebiT advili gamosaTvlelia saZiebeli albaToba:

P{95 X 100}= P{(95 – 90)/15 (X – 90)/15 (100-90)/15}== P{0.33 N(0,1)0.67}= 0.7486 – 0.6293 = 0.1193.

3) SevniSnoT, rom radgan normaluri ganawileba simetrulia

misi saSualos mimarT, am ganawilebis mediana emTxveva saSualos,

anu (6.12)-is ZaliT, , () = (( - )/) = (0) = 0.5.daskvniT statistikaSi gansakuTrebiT xSirad Cven dagvWirdeba

standartuli normaluri ganawilebis p-rigis kvantilis cneba. sazog-adod, uwyveti FX ganawilebis p-rigis kvantili ewodeba argumentisim xp mniSvnelobas, romlisaTvisac

FX(xp) = p. (6.13)

75

standartuli normaluri ganawilebis SemTxvevaSi p-rigis kvan-tils zp-Ti aRniSnaven. am ganawilebis simetrulobis gamo, adgili aqvs

Semdeg Tanafardobas:

zp = - z 1- p. (6.14)

4) damoukidebel normalur SemTxveviT sidideTa nebismieri

wrfivi kombinacia kvlav normaluria: vTqvaT, X1, X2, …, Xn damoukidebe-

li normaluri SemTxveviTi sidideebia da ),(~ 2iii NX . maSin

n

iii

n

iii

n

iii ccNXcZ

1

22

11

,~ , (6.15)

nebismieri c1, c2, …, cn R.

6.3. binomuri da puasonis ganawilebebis aproqsimacia

normaluri ganawilebiT.

normaluri ganawilebis uaRresad mniSvnelovan Tvisebad iTvle-ba is, rom es ganawileba garkveul pirobebSi SeiZleba gamoyenebul iq-

nas, rogorc miaxloeba (aproqsimacia) ama Tu im ganawilebisa, roca am

ukanaskneliT albaTobaTa gamoTvla sakmarisad rTulia (rom aRaraferi

vTqvaT gamoTvlis saerTod SesaZleblobaze). am nawilSi Cven Sevecdeb-

iT warmovaCinoT normaluri ganawilebis es Tviseba binomuri da puas-

onis ganawilebebisaTvis.

6.3.1. binomuri ganawilebebis aproqsimacia normaluriT.

davuSvaT, X SemTxveviTi sidide ganawilebulia binomurad, param-

etrebiT, n da p, amasTan situacia iseTia, rom n da np sakmarisad di-di ricxvebia. maSin gasagebia, rom Zalzed Zneli iqneba

knkknn ppCkSP )1(}{ da

2

1

)1(}{ 21

k

kk

knkknn ppCkSkP

saxis albaTobebis gamoTvla. am albaTobaTa gamosaTvlelad iyeneben

swored normalur aproqsimaciebs:

a)

)1()1(

1)1(}{

pnp

npk

pnpppCkSP knkk

nn ; (6.16)

b)

2

1

)1(}{ 21

k

kk

knkknn ppCkSkP

)1(

5.0

)1(

5.0 12

pnp

npk

pnp

npk, (6.17)

sadac da Sesabamisad aRniSnavs standartuli normaluri ganawi-

lebis simkvrivesa da ganawilebis funqcias.

76

sailustraciod ganvixiloT SemdegimagaliTi 6.3. davuSvaT, albaToba imisa, rom sisxlis TeTri na-

wilaki neutrofiluria (neutrophil) tolia 0.6-is. ras udris albaTo-

ba imisa, rom 100 sisxlis TeTri nawilakidan aseTebis raodenoba aRm-

oCndeba SualedSi [50,75]?amoxsna. binomuri ganawilebiT saZiebeli albaTobis zusti mniS-

vnelobaa

75

50

100100100 )4.0()6.0(}7550{

k

kkkCSP , magram rogorc

vxedavT, is sakmarisad rTuli gamosaTvlelia. (6.17) formulis gamoy-

enebiT ki miviRebT:

4.06.0100

6.01005.49

4.06.0100

6.01005.75}7550{ 100SP

983.0)14.2()16.3( .

6.3.2. puasonis ganawilebebis aproqsimacia normaluriT.iseve, rogorc wina SemTxvevaSi, rodesac puasonis ganawilebis

parametri sakmarisad didi ricxvia, sakmarisad mouxerxebelia

2

1!

}{ 21

k

kk

k

ek

kXkP saxis albaTobebis gamoTvla da aqac iyeneben

normalur aproqsimacias, romelsac Semdegi saxe aqvs:

5.05.0

!}{ 12

21

2

1

kke

kkXkP

k

kk

k

. (6.18)

sailustraciod ganvixiloT Semdegi

magaliTi 6.4. ganvixiloT baqteriaTa raodenoba petris lambaq-

is A areSi. davuSvaT, rom baqteriis dakvirvebis (aRmoCenis) ganawil-

eba eqvemdebareba puasonis kanons parametriT =A, sadac = 0.1 da

A =100sm2. davuSvaT, dakvirvebul iqna 20 baqteria. ramdenad moulod-

nelia es?

amoxsna. pirobis Tanaxmad, gamosaTvlelia P{X 20} albaToba.cxadia, rom saZiebeli albaTobis zusti mniSvnelobaa

20 !

}20{k

k

ek

XP da rogorc 6.3 magaliTSi, is sakmarisad rTu-

li gamosaTvlelia. (6.18) formulis mixedviT ki vpoulobT:

0013.09987.01)3(110

105.0201}20{

XP ,

rac imas niSnavs, rom 20 an meti baqteriis (koloniis) gamoCena

100sm2-ze 10000 lambaqidan SeiZleba mxolod 13-Si moxdes. ra Tqmaunda, es Zalzed iSviaTi xdomilobaa

77

amocanebi1. insultis zusti diagnozis dasma klinikur simptomebze day-

rdnobiT, Zalian Znelia. standartuli diagnosturi testi, romelic

medicinaSi gamoiyeneba insultis dasadgenad damyarebulia angiogramaze.

magram am testis gamoyenebas Tan sdevs garkveuli riski pacientisaTv-

is. amitom SemuSavebul iqna sxvadasxva araagresiuli meTodi, romelicSeiZleba iZleodes iseTsave efeqts, rogorsac angiograma. erT-erTi

aseTi meTodi eyrdnoba tvinis sisxlis nakadis (CBF – cerebral bloodflow) sidides, radgan tipiurad insultian avadmyofebs es sidide Sed-

arebiT daqveiTebuli aqvT. davuSvaT, rom janmrTel populaciaSi CBFganawilebulia normalurad saSualoTi 75 da standartuli gadaxriT

17. iTvleba, rom adamiani insultiania, Tuki CBF-is sidide 40-ze dab-alia. normalur pacientTa ra procenti SeiZleba SecdomiT iqnas miCn-

euli insultiT daavadebulad?

2. glaukoma aris Tvalis daavadeba, romelic xasiaTdeba SidaTv-

alis maRali wneviT. SidaTvalis wnevis sidide Cveulebriv populacia-Si ganawilebulia normalurad saSualoTi 16 mm vwy.sv. da standartu-li gadaxriT 3 mm vwy.sv. normad iTvleba am sididis moxvedra [12, 20]mm vwy.sv. intervalSi. populaciis ra procenti akmayofilebs normis

am kriteriums?

3. rogor Seicvleba pasuxi 6.3 magaliTSi, Tu neutrofilobis

albaToba yoveli nawilakisa iqneboda 0.5, 0.7, 0.9?4. rogor Seicvleba pasuxi 6.4 magaliTSi, Tu A = 50sm2-s? Tu

=0.15, =0.2.5. ras daskvnis gakeTeba SeiZleboda 6.4 magaliTSi, 20-is magi-

vrad rom gamoCeniliyo 15 baqteria, 10 baqteria?6. pirveli Svilis dabadebisas, dedis asaki normalurad aris

ganawilebuli saSualoTi 26.6 da standartuli gadaxriT 3.2 weli.

6.1. ipoveT albaToba imisa, rom SemTxveviT arCeuli aseTi dedis asaki

iqneba 20-ze naklebi;

6.2. ra asakze metia aseTi dedebis asakebis 75%.

7. 40 kviriani Cvilis wona normalurad aris ganawilebuli

3500 gramis toli saSualoTi da 430 grami staandartuli gadaxriT.

7.1 gamoTvaleT im bavSvebis procenti, romelTa wona naklebia 2500gramze;

7.2. ipoveT wonebis I, II da III kvartilebi.

8. axali zelandiis 18 wliani mamakacebis saSualo simaRle 1.7

metria, xolo standartuli gadaxra ki 0.15 metri. ipoveT albaToba

imisa, rom 100 mamakacis saSualo simaRle 1.74 metrze metia.

9. qveviT moyvanilia 1998 wels aSS-Si qmris asakis ganawileba:

78

asaki 24 25-34 35-44 45-54 55-59 60-61

fard. sixS. 0.023 0.178 0.257 0.217 0.079 0.029

asaki 62-64 65-69 70-74 75-79 80-84 85fard. sixS. 0.040 0.062 0.049 0.039 0.019 0.008

ipoveT albaToba imisa, rom SemTxveviT SerCeul ojaxSi: a) qmris asa-

ki ar aRemateba 44 wels; b) qmari 75 wlis an ufro xnieri iqneba.

10. V ml tbis wyalSi organuli nawilakebis raodenoba emorCi-

leba puasonis ganawilebis kanons saSualoTi 0.2V . gamoTvaleT alba-

Toba imisa, rom: a). 50 ml tbis wyali Seicavs 8-ze nakleb organulnawilaks; b). 30 ml tbis wyali Seicavs 2-ze met organul nawilaks;

g). 10 ml tbis wyali Seicavs zustad 3 organul nawilaks.

11. sakonditro fabrika amzadebs didi raodenobiT ferad tkbi-

leuls. cnobilia, rom saSualod tkbileulis 20% mwvane ferisaa.

SekvraSi aris 20 tkbileuli. igulisxmeT, rom SekvraSi tkbileuli

xvdeba SemTxveviT fabrikis mier damzadebuli mTeli produqciidan da

gamoTvaleT albaToba imisa, rom SekvraSi zustad 7 tkbileuli iqneba

mwvane.

12. mefrinveleobis fabrikaSi warmoebuli 6 – 6 kvercxi Calag-

ebulia yuTebSi. TiTeuli kvercxis gatexvis albaToba sxva kvercxebi-sagan damoukideblad aris 0.1. yuTs davarqvaT cudi, Tu masSi devs

sul cota 2 gatexili kvercxi. ipoveT albaToba imisa, rom SemTxvev-

iT SerCeuli yuTi cudia.

13. garkveul tropikul kunZulze romelime konkretul TveSi

qariSxalis moxdenis albaTobaa 0.08. gamoiyeneT binomialuri ganawil-

eba da gamoTvaleT albaToba imisa, rom wlis ganmavlobaSi qariSxali

iqneba orze met TveSi.

14. saxlis mepatrones surs baRSi daTesos balaxi. Teslis mob-

neva xdeba SemTxveviT da baRis konkretul 1 kv. santimetri farTobis

mqone nawilSi davardnili Teslis raodenoba warmoadgens puasoniskanoniT ganawilebul SemTxveviT sidides, romlis saSualo baRis nawi-

lis farTobis proporciulia. baRis farTobia 50 kv. metri da iTeseba610 balaxis Tesli. gamoTvaleT: a). -s saSualo; b). albaToba imisa,

rom 1 kv. sm. farTobis mqone nawilSi ar daecema arcerTi Tesli; g).

{ 0 4}P an .

15. kompiuteris “kartrijis” muSaobis xangrZlivobaa saaTi.

SemTxveviTi sididis ganawilebis simkvrivea:2 , 4;

( )0,

cx xf x

Tu

sxvagan.

gamoTvaleT c mudmivis mniSvneloba. ipoveT albaToba imisa, rom: a).

“kartriji” imuSavebs sul cota 500 saaTi; b). “kartriji” Sesacvle-

79

li gaxdeba manam sanam is imuSavebs 600 saaTi; g). ori “kartriji” Se-sacvleli iqneba manam sanam TiToeuli imuSavebs 600-600 saaTi; d).

oTxi “kartrijidan” ori imuSavebs 600 saaTze mets, xolo ori 600

saaTze naklebs.

III. daskvniTi statistika

l e q c i a 6.

Tavi 7. SefasebaTa Teoria. wertilovani Sefasebebi.

7.1. Sesavali.

wina leqciebSi Cven vixilavdiT iseT amocanebs, romlebSic moc-

emuli iyo populaciis ganawilebis parametrebi da amocana mdgomareo-

bda martivi xdomilobebis albaTobaTa saSualebiT, modelis specifik-

aze dayrdnobiT, gamogveTvala SedarebiT rTuli xdomilobebis albaT-

obebi, sxva sityvebiT rom vTqvaT, amocana mdgomareobda populaciis

parametrebze dayrdnobiT SerCevis yofaqcevis “ganWvretaSi”. danarCenileqciebi ganekuTvneba daskvniTi statistikis Seswavlas, romlis mTav-

ari amocanac rogorc gvaxsovs, mdgomareobs SerCevidan miRebuli inf-

ormaciis safuZvelze daskvnebis ganzogadebaSi mTeli populaciisaTvis.

amdenad, darCenil leqciebSi Sesaswavli amocanebi ukve Seswavlili

amocanebis Sebrunebulia. sqematurad, sakiTxi ase dgas: SerCevidan populaciisaken. pirvel ori leqcia, sadac ganxiluli iyo aRweriTi

statistikis amocanebi, garkveuli azriT SeiZleba CaiTvalos am sqema-turi gadasvlisaTvis mosamzadebel etapad. albaTobis Teoriis roli

(leqciebi 3-5), ki imaSi mdgomareobs, rom Teoriulad daasabuTos es

sqematuri gadasvla. mTavari momenti am nawilebis Serwymaa, rac imas

niSnavs, rom axla ukve SerCevas Cven unda vuyuroT ara mxolod rog-

orc ricxvebis raRac (calsaxad gansazRvrul) erTobliobas, aramed

misi elementebi (calkeuli ricxviTi monacemi) ganxilul unda iqnas,

rogorc romeliRac SemTxveviTi sididis konkretuli realizacia, Sem-

TxveviTi sididisa, romelsac Tavisi ganawileba aqvs, SemTxveviTi sid-

idisa, romelsac Tavisi ganawilebis parametrebi aqvs da a.S. maSasada-

me, ricxviTi monacemi aRaraa mxolod ricxvi: mis ukan dgas SemTxvevi-

80

Ti sidide Tavisi ganawilebis kanoniTa da parametrebiT. mxolod aseTimidgomiT SeiZleba raime azriani daskvnebis keTeba ama Tu im movlenis

Sesaxeb movlenaTa masobrivi (bevrjeradi) gameorebis SemTxvevaSi.

7.2. wertilovani Sefasebebi.

daviwyoT magaliTebis ganxilviT.

magaliTi 7.1. davuSvaT, Cven vuzomavT sistoluri sisxlis wnev-

is sidides samoas soflis mosaxleobas, romlis Sesaxebac wina dakvi-

rvebebi ar arsebobs, Tumca, Cveni rwmeniT es sidide normaluradaa ga-

nawilebuli, magram ucnobi (, 2) parametrebiT. rogor SeiZleba Sef-asdes es parametrebi?

magaliTi 7.2. davuSvaT, romelime adamianis organizmSi sisxlis

TeTr nawilakebSi neitrofilurebis raodenoba binomuradaa ganawileb-

uli, magram CvenTvis ucnobia ganawilebis parametri p. rogor Sevafas-oT is?

magaliTi 7.3. davuSvaT, baqteriaTa koloniebis raodenoba petr-

is lambaqis garkveul areze ganawilebulia puasonis kanoniT, magram

CvenTvis ucnobia ganawilebis parametris mniSvneloba. rogor Sevaf-asoT es parametri?

rogorc vxedavT, samive SemTxvevaSi cnobilia ganawilebis saxe,

magram ucnobia maTSi Semavali parametrebi. amocana ki mdgomareobs ga-

nawilebaTa am parametrebis SefasebaSi. ra unda gvesmodes saerTod

“Sefasebis” qveS? amis gasaazreblad, gavixsenoT yvelasaTvis kargad

cnobili iracionaluri ricxvi, romelic wrewiris sigrZis mis dia-

metrTan Sefardebis tolia da davsvaT aseTi SekiTxva: ras udris -sricxviTi mniSvneloba? alabaT, uamravi Cvengani ityvis, rom es aris

3.14. magram cxadia, rom es pasuxi swori ar aris, radgan 3.14 aris ricxvis mxolod miaxloebiTi mniSvneloba measedamde sizustiT. aseve,

3.142 aris -s miaxloebiTi mniSvneloba meaTasedamde sizustiT da a.S.

aseT SemTxvevebSi vityviT, rom Cven vasaxelebT ricxvis zusti mniS-vnelobis sxvadasxva Sefasebebs. es Sefasebebi ki, imiT xasiaTdeba, rom

| -3.14| < 0.01, | -3.142| < 0.001 da a.S. analogiurad, ama Tu im ganawil-ebis parametris Sefasebis qveS unda gvesmodes sidide, romelic param-

etris “WeSmarit” mniSvnelobasTan “sakmarisad axlos” iqneba. swored

aseT Sefasebebs uwodeben ganawilebaTa parametrebis wertilovani Sef-

asebebi. magram wina SemTxvevisagan gansxvavebiT (roca cnobili ricx-

via), am SemTxvevaSi parametris “WeSmariti” mniSvneloba ucnobi sidid-

ea. maS rogor unda davaxasiaToT Sefasebis siaxlove “WeSmarit” mniS-vnelobasTan? swored am sakiTxs eZRvneba es nawili.

davuSvaT, martivi SerCeviTi meTodis safuZvelze miRebulia nmoculobis SerCeva raime generaluri erTobliobidan (gavixsenoT I

81

leqcia): x1, x2,..., xn. Sefasebis amocanis dasasmelad, rogorc zemoT aR-vniSneT, Cven am ricxvebs unda vuyuroT ara mxolod ricxvebs, rog-

orc aseTs, aramed, garkveuli SemTxveviTi sidideebis realizaciebs,

anu unda davuSvaT, rom arsebobs damoukidebeli erTnairad ganawileb-

uli SemTxveviTi sidideebis mimdevroba X1, X2,..., Xn, romelTa “konkre-

tuli gaTamaSebis” Sedegadac miviReT swored x1, x2,..., xn ricxvebi. ma-

Sasadame, generaluri erTobliobidan SerCevis amokrefis proceduras

Cven vaigivebT damoukidebeli, erTnairad ganawilebuli X1, X2,..., Xn Se-

mTxveviTi sidideebis gaTamaSebis procedurasTan an rac igivea, mas va-

igivebT im saerTo ganawilebis mqone X SemTxveviTi sididis n jeradi

damoukidebeli gaTamaSebis procedurasTan, rogoradac ganawilebuliaX1, X2,..., Xn SemTxveviTi sidideebi. rogorc es zemoT moyvanili magal-

iTebidan davinaxeT, es saerTo ganawileba (anu populaciis ganawileba)

SeiZleba iyos normaluri, binomuri, puasonis da sxva mravali ganawi-

leba. aseT SemTxvevebSi amboben, rom x1, x2,..., xn warmoadgens martiv Se-mTxveviT SerCevas (realizacias) Sesabamisad, normaluri, binomuri, pu-

asonis da a.S. generaluri erTobliobidan.

gavixsenoT monacemTa ZiriTadi ricxviTi maxasiaTeblebi me-2

leqciidan. eseni iyo SerCeviTi saSualo da SerCeviTi dispersia:

nx =

n

iix

n 1

1da 2

ns =

n

ini xx

n 1

2)(1

1. (7.1)

radgan x1, x2,..., xn ricxvebs ganvixilavT rogorc X1, X2,..., Xn SemTxvev-

iTi sidideebis gaTamaSebis Sedegebs, bunebrivia, SerCeviTi saSualo da

SerCeviT dispersiac ganvixiloT rogorc Sesabamisad

nX =

n

iiX

n 1

1da 2

nS =

n

ini XX

n 1

2)(1

1(7.2)

SemTxveviTi sidideebis gaTamaSebis Sedegebi (realizaciebi). sailust-

raciod moviyvanoT Semdegi

magaliTi7.4. cxrilSi mocemulia n = 10 moculobis xuTi SerCe-

va: dakvirvebebi axalSobilTa wonebze unciebSi (1unc.=28.35g) da Sesa-bamisi saSualoebi da standartuli gadaxrebi:

SerCeva

individi 1 2 3 4 51 97 177 97 101 1372 117 198 125 114 1183 140 107 62 79 784 78 99 120 120 1295 99 104 132 115 876 148 121 135 117 1107 108 148 118 106 1068 135 133 137 86 116

82

9 126 126 126 110 14010 121 115 118 119 98

10x 116.90 132.80 117.00 106.70 111.90

10s 21.70 32.62 22.44 14.13 20.46

SeviswavloT (7.2) formulebiT ganmartebuli SemTxveviTi sidi-

deebis albaTuri Tvisebebi, pirvel rigSi ki - maTi ricxviTi maxasiaT-eblebi. daviwyoT nX -is ganxilviT.

7.2.1. populaciis saSualos wertilovani Sefaseba.

maTematikuri lodinisa da damoukidebel SemTxveviT sidideTa

jamis dispersiis Tvisebebis gaTvaliswinebiT miviRebT, rom

E nX = EXEXn

n

ii

1

1= da D nX =

nDXn

nDX

n

n

ii

2

21

2

11

, (7.3)

saidanac vxedavT, rom nX -is saSualo (maTematikuri lodini) emTxveva

populaciis ucnob parametrs, xolo dispersia 2/n ki n -is zrda-

sTan erTad miiswrafis nulisaken, rac imaze metyvelebs, rom nX Sem-

TxveviTi sidide saSualod “axlosaa” ucnobi parametris ricxviT

mniSvnelobasTan. maSasadame, misi konkretuli realizaciis ricxviTi

mniSvnelobac nx =

n

iix

n 1

1 axlos unda iyos parametris imave mniSvne-

lobasTan. es magaliTi naTlad gviCvenebs ucnobi parametrisaTvis Sef-

asebis agebis gzas: ganawilebis ucnobi parametris Sinaarsidan gamomd-

inare (maTematikuri lodinia is, dispersia, mediana, moda, Tu kidevsxva rame), avagoT misi SerCeviTi maxasiaTebeli da am ukanasknelSi,

rogorc SerCeviTi mniSvnelobebis funqciaSi CavsvaT x1, x2,..., xn -is Se-

sabamisi X1, X2,..., Xn SemTxveviTi sidideebi; miRebul SemTxveviT sidid-

es vuwodoT ucnobi parametris Sefaseba, xolo mis realizacias konk-

retuli gaTamaSebisaTvis – ucnobi parametris miaxloebiTi mniSvnel-oba.

sabolood, Sefasebis sqema aseTnairad gamoiyureba:a) mocemulia ricxviTi monacemebi x1, x2,..., xn ;

b) Cven maT SevusabamebT damoukidebel erTnairad ganawilebul

X1, X2,..., Xn SemTxveviTi sidideebs, romelTa ganawilebis funqcia

FX(x;) (an ganawilebis simkvrive fX(x;)) damokidebulia ucnob param-

etrze (populaciis parametrze);

g) vagebT parametris SerCeviT analogs, anu x1, x2,..., xn argumen-

tebis raime funqcias T(x1, x2,..., xn) da ucnobi parametris Sefasebad

vacxadebT SemTxveviT sidides n~

T(X1, X2,..., Xn).

83

davsvaT axla aseTi SekiTxva: rogori unda iyos T funqcia, rom

n~ aRmoCndes parametris “kargi” Sefaseba?

rogorc SerCeviTi saSualos magaliTma dagvanaxa, erT-erTi “ka-

rgi Tviseba” aris Sefasebis e.w. gadauadgilebadobis (Caunacvleblob-

is) Tviseba.

gansazRvra 7.1. ucnobi parametris n~

Sefasebas ewodeba gada-

uadgilebadi anu Caunacvlebeli, Tu:

E n~

= , an rac igivea, E( n~

- ) = 0, (7.4)

sxva sityvebiT, rom vTqvaT, saSualod zustad unda vafasebdeT Sesaf-

asebel parametrs da Sefasebis dros ar unda vuSvebdeT “sistematur”

Secdomebs.

SefasebaTa meore aranakleb mniSvnelovani Tviseba aris e.w. Zal-

debulobis (Zalmosilebis) Tviseba.

gansazRvra 7.2. ucnobi parametris n~ Sefasebas ewodeba Zal-

debuli anu Zalmosili, Tu nebismieri 0 ricxvisaTvis:

didiaroca nP n ,0}|~

{| . (7.5)

Sinaarsobrivad, Zaldebuloba niSnavs imas, rom rac ufro meti

monacemiT xdeba ucnobi parametris Sefaseba, miT ufro naklebia Sansi

Sefaseba gadaixaros Sesafasebeli sidididan ragind mcire -ze ufrometad, anu Sefaseba TandaTan stabilurdeba. gavixsenoT am situaciis

Sesabamisi 4.1 magaliTi aSS-Si axalSobilebis Sesaxeb:

kalendarulidro

dabadebulibiWebis raod.

axalSobilTasaerTo raod.

fardobiTisixSire

1965 1 927 054 3 760 358 0.512471965-1969 9 219 202 17 989 361 0.512481965-1974 17 857 857 34 832 051 0.51268

Tu Sesafasebel parametrad miviCnevT biWis dabadebis p albaT-

obas, maSin axalSobilTa saerTo raodenobis zrdasTan erTad, biWebis

raodenoba ise izrdeba, rom fardobiTi sixSire stabiluri xdeba da

adgili aqvs e.w. did ricxvTa kanons:

( )0,

n AP p n

n

roca didia , (7.6)

sadac A aRniSnavs biWis dabadebis xdomilobas, xolo – n(A) biWebis

raodenobaa n axalSobils Soris.

davubrundeT, gadauadgilebad Sefasebebs da davsvaT aseTi Seki-Txva: ramdenad arsebiTia Sefasebis gadauadgilebadoba? sakiTxis ase

dasma imiTaa ganpirobebuli, rom populaciis parametris

nX =

n

iiX

n 1

1Sefasebis msgavsad, gadauadgilebadia magaliTad, yvela

84

e.w. Sewonili, anu

n

iiin Xc

1

~ saxis Sefasebac, romelTaTvisac

11

n

iic . marTlac,

n

ii

n

iiin cEXcE

11

~. maSin ratom vamjob-

ineT yvela aseT Sefasebas iseTi, romlisaTvisac c1 = c2 =…= cn =1/n?saqme imaSia, rom Sefasebis standartuli Secdoma, romelic sazoga-

dod ganimarteba, rogorc nD~ , nX -s aqvs minimaluri

n

iiin Xc

1

~

saxis Sefasebebs Soris; kerZod, rogorc vnaxeT, D nX = 2/n, anu

nXD n / , maSin, roca

n

iin cD

1

2~ da 11

n

iic pirobidan

ki gamomdinareobs, romn

cn

ii

1

1

2

. ase, rom Cven mivadeqiT Sefasebe-

bis “sikargis” yvelaze mZlavr kriteriums – SefasebaTa efeqturobisTvisebas.

gansazRvra 7.3. gadauadgilebad SefasebaTa klasSi minimaluri

standartuli Secdomis mqone Sefasebebs efeqtur Sefasebebs uwodeben.

SevniSnoT, rom populaciis standartuli gadaxra da saSualos

standartuli Secdoma sxvadasxva obieqtia: pirveli maTgani axasiaTebs

X, xolo meore ki nX SemTxveviT sidides. rodesac populaciis stan-

dartuli gadaxra ucnobia (rac praqtikaSi arc Tu iSviaTia), saSu-alos standartul Secdomas n/ -s afaseben nSn / sididiT.

magaliTi 7.5. gamovTvaloT saSualos standartuli Secdomis

Sefaseba 7.4 magaliTSi moyvanili me-3 SerCevisaTvis. gvaqvs:

09.710/44.2210/10 s .

mivyveT wertilovani Sefasebebis agebis zemoT moyvanil gzas da

vnaxoT rogoria ama Tu im populaciis Sesabamisi parametris wertil-

ovani Sefasebebi.

7.2.2 populaciis dispersiis wertilovani Sefaseba.

ganvixiloT raime populacia, romelsac gaaCnia 2 dispersia.

rogorc me-2 leqciidan gvaxsovs, 2-is empiriuli analogi iyo 2ns .

amitom populaciis dispersiis wertilovan Sefasebad bunebrivia agve-

Ro 2nS =

n

ini XX

n 1

2)(1

. magram mas rogorc dispersiis Sefasebas, Cven

vamjobineT sxva wertilovani Sefaseba, kerZod, 2nS =

85

n

ini XX

n 1

2)(1

1sidide. riTaa es ganpirobebuli? saqme imaSia, rom

2nS warmoadgens populaciis dispersiis gadauadgilebad Sefasebas, 2

nS

ki ara. marTlac, gamovTvaloT 2nS -isa da 2

nS -is maTematikuri lodine-

bi. gvaqvs:

E 2nS = 2 2 2

1 1

1 1( )

1 1

n n

i n i ni i

E X X EX n EXn n

22 2 2 2 2 2 2( )

1 1n

n nEX

n n n

.

maSasadame, sabolood,

E 2nS = 2 . (7.7)

anlogiurad gamoiTvleba, rom

E 2nS = 2 - 2/n. (7.8)

Sesabamisad, 7.1 gansazRvris Tanaxmad, 2nS warmoadgens populac-

iis 2 dispersiis gadauadgilebad Sefasebas, xolo 2nS -s, rogorc 2-

is wertilovan Sefasebas, gaaCnia e.w. Canacvleba, romelic 2/n-istolia, rac marTalia umniSvneloa didi moculobis SerCevebisaTvis,

magram mniSvnelovania SerCevebis patara moculobebis dros.

7.3. centraluri zRvariTi Teorema.aqamde Cven araferi gviTqvams populaciis ganawilebis konkret-

uli saxis Sesaxeb. Tu X ~ N(, 2), maSin rogorc normaluri SemTx-

veviTi sidideebis wrfivi kombinacia (gaixseneT me-5 leqcia) nX Sem-

TxveviT sididec ganawilebulia normalurad da rogorc (7.3) Tanaf-

ardobidan Cans, nX ~ N(, 2/n). magram rogoria nX SemTxveviT sidid-

is ganawileba, roca X araa normalurad ganawilebuli SemTxveviTi si-

dide? am kiTxvaze pasuxs iZleva albaTobis Teoriis erT-erTi umniSv-

nelovanesi Teorema, romelsac saxelad hqvia

centraluri zRvariTi Teorema: Tu X1, X2,..., Xn damoukidebeli

erTnairad ganawilebuli SemTxveviTi sidideebia saSualoTi da dis-

persiiT 2, maSin nebismieri a da b ricxvebisaTvis, a<b:

didia.roca nabbn

XaP n ),()(

/

(7.9)

rogorc vxedavT, damoukidebel SemTxveviT sidideTa saSualo

ariTmetikulis asimptoturi ganawilebisaTvis araviTari mniSvneloba

ara aqvs Sesakrebebis ganawilebis kanons, mTavaria, rom maT gaaCndes

sasruli dispersia da aseT pirobebSi saSualo ariTmetikulis asimpt-

86

oturi ganawileba normaluria. am Teoremis gamoyenebis sailustraci-od ganvixiloT Semdegi

magaliTi 7.6. 1000 axalSobilis saSualo wona aRmoCnda 112 un-cia, xolo standartuli gadaxra 20.6 uncia. ras udris albaToba imi-

sa, rom SemTxveviT arCeuli 10 axalSobilis saSualo wona aRmoCndeba

98-dan 126 unciamde?

amoxsna. pirobis Tanaxmad, davaskvniT, rom = 112, n/ =

20.6/ 10 = 6.51. amitom centraluri zRvariTi Teoremis gamoyenebiT,

saZiebeli albaToba tolia

51.6

112126

51.6

112

51.6

11298}12698{ 10

10

XPXP

.968.01984279.021)15.2(2)15.2()15.2(

7.4. binomuri populaciis p parametris wertilovani Sefaseba.

daviwyoT magaliTis ganxilviT.

magaliTi 7.7. Cven gvainteresebs avTvisebiani simsivniT daavadeb-

is done 45-54 ww. asakobrivi gjgufis qalebisaTvis. SemTxveviTi SerC-

evis meTodiT SerCeul iqna am asakis 5000 qali, romelTa Soris 28-saRmoaCnda es daavadeba. binomurad ganawilebuli SemTxveviTi sididis

Semotanis dros Cven ar gviTqvams, Tu rogor SeiZleba am sididis fo-

rmireba. dakvirvebis yoveli obieqtisaTvis (am SemTxvevaSi qalisaTvis),

SemoviRoT ormniSvnelobiani diskretuli SemTxveviTi sidide, romel-

ic Rebulobs mniSvnelobebs 1-sa (qali daavadebulia) da 0-s (qali

araa daavadebuli), albaTobebiT p da 1-p, Sesabamisad:Xi 0 1P 1-p p

sazogadod, aseT SemTxveviT sidideebs bernulis SemTxveviT si-dideebs uwodeben. SevniSnoT, rom EXi = p da DXi = p(1-p).

advili saCvenebelia, rom

n

iiXX

1

(Cvens SemTxvevaSi daavadebu-

lTa saerTo raodenoba) SemTxveviT sidides aqvs binomuri ganawileba,parametrebiT n da p. sxva sityvebiT rom vTqvaT, binomuri SemTxveviTisidide warmodgeba rogorc jami damoukidebeli, erTnairad (bernulis

kanoniT) ganawilebuli SemTxveviTi sidideebisa. amitomacaa, rom

nppEXEXn

i

n

ii

11

da )1()1(11

pnpppDXDXn

i

n

ii

.

pirveli tolobidan davaskvniT, rom

n

iiX

nEp

1

1, anu p pa-

rametrs aqvs garkveuli (bernulis) SemTxveviTi sidideebis saSualo

87

mniSvnelobis azri. amitom 7.2.1. punqtis msgavsad, p parametris wert-ilovan Sefasebad iReben sidides:

n

XX

np

n

iin

1

1~ . (7.10)

radgann

pppD n

)1(~ , am Sefasebis standartuli Secdoma cxa-

dia, tolia npp /)1( sididisa.

maSasadame, sabolood, Tu X aris binomurad ganawilebuli SemT-

xveviTi sidide parametrebiT p da n, p parametris gadauadgilebad we-

rtilovan Sefasebas warmoadgens SerCevis proporcia np~ X / n da mi-

si standartuli Secdomaa npp /)1( .

Cveni magaliTisaTvis, X = 28, n = 5000, amitom np~ 28/5000 =

0.0056 standartuli Secdoma ki, tolia 0011.05000/9944.00056.0 .

7.5. puasonis populaciis parametris wertilovani Sefaseba.me-4 leqciidan gavixsenoT, rom X SemTxveviT sidides aqvs puas-

onis ganawileba, Tu ek

kXPk

!}{ , sadac = t. garda amisa,

gavixsenoT, rom EX = = t, sadac t aRniSnavs dros, xolo kidrois erTeulSi momxdar (iSviaT) xdomilobaTa raodenobas. 5.4 maga-

liTSi cnobili iyo -s mniSvneloba da misi saSualebiT viTvlidiTgarkveul albaTobebs. axla amocana piriqiTaa, mocemulia X SemTxvevi-

Ti sididis mniSvneloba. rogor SevafasoT -s mniSvneloba? cxadia,

rom EX = t, tolobidan = EX / t = E(X / t) da maSasadame, Tu Cven avi-

RebT Sefasebis rolSi ~ = X / t, maSin E~ = da ~ aRmoCndeba para-metris gadauadgilebadi wertilovani Sefaseba. sailustraciod ganvi-

xiloT Semdegi

magaliTi 7.8. 1970 wels voburnSi (masaCusetsis Stati) aRiri-

cxa 12 leikemiis SemTxveva 19 wlamde mozardTa Soris. CavTvaloT,

rom 10 wlis ganmavlobaSi (1970-1979ww) mozardTa raodenoba iyo

mudmivi da iyo 12000-is toli. rogoria parametris wertilovani

Sefaseba?amoxsna. cxadia, rom 10 wlis ganmavlobaSi dagrovebul SemTxv-

evaTa saerTo raodenoba tolia 1200010 = 120000. amitom

~ = X/t = 12 /120000 = 0.0001.amocanebi

1. monacemTa ori sxvadasxva populaciidan iReben erTnairi moc-

ulobis or SerCevas. Ppirveli populaciis standartuli gadaxraa

17.0, xolo meoris ki 30.0. rogor fiqrobT, romeli populaciidan

88

aRebuli SerCevis standartuli Secdoma iqneba meti. axseniT Tqvenipasuxi.

2. populaciidan Rebulia ori SerCeva. Ppirveli SerCevis mocu-

lobaa 24, xolo meoris ki 10. rogor fiqrobT, romeli SerCevis saS-

ualo iqneba populaciis saSualosTan ufro axlos? ratom?

3. SegrZnebis Ziebis skalis mixedviT maRali qulis mqone adami-

anebs male bezrdebaT dawyebuli aqtivobebi da Zalian esaWiroebaT ax-

al da zogjer riskian saqmianobebSi CarTva. skalaze qulebi 10-dan

40-mde meryeobs. Aam skalaze 5 SerCeva gazomes. SerCevebi sxvadasxva

moculobis iyo.

SerCeva 1 2 3 4 5

moculoba 4 9 16 25 100

3.1 TiToeuli SerCevisaTvis gamoTvaleT SerCevis standartuli Secd-

oma;

3.2 ipoveT albaToba imisa, rom SerCevis saSualo 25-is toli an mas-

ze meti iqneba;3.3. standartuli Secdomis da moculobis ra kavSiris ilustraciaa

es magaliTi.

4. qalaqis mosaxleobis 75%-is asaki 20 welze metia. ipoveT

albaToba imisa, rom 200 kacian SerCevaSi X metia 140-ze da naklebia

180-ze, sadac X ari 20 wels gadacilebuli adamianebis raodenoba,

5. nebismieri mocemuli wlisaTvis, im adamianebis proporcia,

romlebic dadian eqimTan aris 81%. Aaxalgazrdebis 78 kacian SerCev-

aSi, 56-ma aRniSna rom iyo misuli eqimTan bolo wlis ganmavlobaSi.

gansazRreT SerCevis jgufis warmomadgenlebi ufro naklebad dadian

Tu ara eqims, vidre mTeli populaciis warmomadgenlebi.6. maRaroSi momuSave muSebis 30%-s aRmoaCnda sxvadasxva sunT-

qviTi problemebi. am muSebis asaki ZiriTadad iyo 50 weli. gamoTval-

eT, maRaroSi momuSave muSebis 80 kacian SemTxveviT SerCevaSi, ras ud-

ris albaToba imisa, rom 35 muSas eqneba sunTqvasTan dakavSirebuli

problemebi.

7. supermarketSi 200 naxevarStatSi myofi 18 wlis TanamSrom-

lebis saaTobrivi anazRaureba saSualod aris 9.50 dolari, xolo

standartuli gadaxra 0.5 dolari. iTvleba, rom am qveyanaSi zogadad

18 wlis naxevarStatiani TanamSromlebi saSualod iReben 10.30 dola-

ris odenobis saaTobriv anazRaurebas. Seesabameba Tu ara es Tvalsazr-isi 200 kacian SerCevis Sedegs.

8. erT-erTi sadazRveo kompania dapirda Tavis momxmareblebs,

rom miRebuli saCivrebis 90%-s gadaWrida 30 dRis ganmavlobaSi. Se-

mTxveviT aRebul 75 saCivrian SerCevaSi 55 gadawyda 30 dReSi. arseb-

obs momxmareblis erTi jgufi, romelic gamoTqvams seriozul eWvs

kompaniis mier dasaxelebul saCivrebis odenobasTan dakavSirebiT. maTi

89

azriT sinamdvileSi, saCivrebis gacilebiT ufro naklebi proporciisganxilva xdeba 30 dReSi. Mmomxmareblebis da kompaniis mosazrebidan,

romels uWerT yvelaze ufro mxars?

9. erT erTi marketis yoveldRiuri saSualo Semosavali 1000

laria, standartuli gadaxra ki 650 lari. ras udris albaToba imisa,

rom Semosavali iqneba 1800-ze meti?

10. maRaziis mepatrone, axdens misi satvirTo manqanis mier mox-

marebuli sawvavis monitorings. qvemoT mocemuli monacemebi, aRweren

manqanis mier gavlili kilometrebis raodenobas TiToeul daxarjul

litrze (km/l):11.6 10.7 11.5 10.1 9.6 10.9 11.0 10.6 8.5 9.7 9.3 8.6 8.2 10.6

10.8 9.4 8.5 10.1 9.6 11.1 11.6 8.5 11.6 9.3 10.3 10.2 10.5.

saSualod ramden kilometrs gadis aseTi tipis satvirTo manqana TiT-

oeul daxajul litrze?

11. gamoTvaleT evropeli moqalaqis wliuri Semosavlis SerCev-

iTi saSualo da Sesworebuli standartuli gadaxra, Tu qvemoT moyva-

nilia SemTxveviT SerCeuli 50 evropeli moqalaqis Semosavlebi gazo-mili aTasobiT evroebSi:

84 14 31 72 26 49 252 104 31 8

3 18 72 23 55 133 16 29 225 138

85 24 391 72 158 4340 346 19 5 846

461 254 125 61 123 60 29 10 366 47

28 254 6 77 21 97 6 17 8 82

l e q c i a 7.

Tavi 8. SefasebaTa Teoria. intervaluri Sefasebebi.

8.1. Sesavali.

xSirad populaciis saSualos zusti mniSvnelobis magivrad, sa-

Wiroa misi sarwmuno (ndobis) intervalis ageba. Tu modeli normalu-

ria, maSin amis gakeTeba SesaZlebelia zustad, xolo winaaRmdeg SemTx-

vevaSi centralur zRvariT Teoremaze dayrdnobiT, SesaZlebelia miax-

loebiTi (asimptoturi) ndobis intervalis ageba. gavixsenoT 7.4 maga-

liTi. Tu Cven daveyrdnobiT pirvel SerCevas, maSin saSualos werti-

lovani Sefaseba iqneba 10x = 116.9 uncia. Tumca es Sefaseba mocemuli

SerCevis safuZvelze saukeTesoa, Cven mainc ver viqnebiT darwmunebuli

imaSi, rom = 116.9, Tundac imitom, rom meore aseTive (imave cxril-

idan) SerCeva iZleva sul sxva Sedegs: 10x = 132.8 uncias, mesame – kid-

ev sxvas da a.S. aqedan Cans, rom aucilebelia -saTvis garkveuli int-

90

ervalis ageba, romelSic is didi albaTobiT moxvdeba. aseT interval-ebs ndobis intervalebs uwodeben da maTi ageba SesaZlebelia rogorc

vTqviT, centraluri zRvariTi Teoremis gamoyenebiT.

8.2. populaciis saSualos intervaluri Sefasebebi.

ucnobi populaciis saSualos intervaluri Sefasebis ageba dam-

okidebulia imaze, cnobilia Tu ara populaciis standartuli gadaxr-

is ricxviTi mniSvneloba. amitom intervaluri Sefasebis agebis amocan-

as populaciis saSualosaTvis Cven or nawilad gavyofT: 1) roca parametris ricxviTi mniSvneloba cnobilia da 2) roca parametrisricxviTi mniSvneloba ucnobia.

8.2.1. saSualos intervaluri Sefaseba cnobili -s dros.

davuSvaT, populaciis parametri cnobilia. maSin centraluri

zRvariTi Teoremis Tanaxmad, nX ~N(,2/n), anu ( ) / ~nn X (0,1)N . normaluri ganawilebis kvantilebis cxrilebidan Cven SegviZ-

lia yoveli p ndobis albaTobis Sesabamisi zp kvantilis moZebna, aseve

iseTi up ricxvis moZebna, rom

puXnP pn /)( , anu

pnuXnuXP pnpn .

p ndobis albaTobis mqone anu 100 p%-ian ndobis intervals uc-

nobi saSualosaTvis aqvs Semdegi saxe:

nuxnux pnpn // . (8.1)

advili misaxvedria kavSiri zp da up sidideebs Soris:

up = z(1+p)/2. (8.2)

ra Tqma unda, sasurvelia ndobis p albaToba iyos rac SeiZlebadidi, magram “samwuxarod” is ver iqneba 1-is toli, radgan Sesabamisi

ndobis intervali aRmoCndeba (-;+)-is toli da es ki arafris

mTqmelia populaciis saSualoze. amitom iqcevian Semdegnairad: afiqsi-

reben raime mcire -s, magaliTad, = 0.05 an = 0.01 da iReben p = 1 - . normaluri ganawilebis cxrilebidan vpoulobT, rom u0.95 = z0.975 ==1.96 da u0.99 = z0.995 = 2.58. Sesabamisad, (8.1) Tanafardobidan gamomdina-re, 95%-ian ndobis intervals aqvs saxe:

nxnx nn /96.1/96.1 , (8.3)

xolo 99%-iani ndobis intervali ki iqneba:

nxnx nn /58.2/58.2 . (8.4)

SevniSnoT, rom ndobis albaTobis zrda iwvevs ndobis interva-lis gafarToebas, rac bunebrivia.

91

davubrundeT 7.4 magaliTs, sadac = 20.6. maSin n = 10-saTvis,magaliTad, 95%-iani ndobis intervali ase gamoiyureba:

[ 10x -12.8, 10x +12.8],

xolo cxrilSi mocemuli xuTi SerCevisaTvis, gveqneba:

10x 116.90 132.80 117.00 106.70 111.90

inter. [104.1,129.7] [120,145.6] [104.2,129.8] [93.9,119.5] [99.1,124.7]

gavixsenoT, rom 7.6 magaliTidan 1000 axalSobilis saSualo

wona aRmoCnda 112 uncia da -s Sefasebad aviReT = 112. rogorcvxedavT, -s es mniSvneloba yvela intervalSia, garda meorisa. radaskvnis gakeTeba SeiZleba aqedan? Cven ver vityviT, rom 95%-iani al-

abaTobiT ucnobi moxvdeba romelime konkretul intervalSi, magram

risi Tqmac SegviZlia, is aris, rom yvela aseTi ndobis intervalebis

95% moicavs -s.

8.2.2. saSualos intervaluri Sefaseba ucnobi -s dros.

davuSvaT, axla, rom parametris ricxviTi mniSvneloba ucno-

bia. maSin rogorc wina leqciaSi aRvniSneT, saSualos standartul

Secdomas n/ -s afaseben nSn / sididiT. magram es sidide n/ -

sagan gansxvavebiT SemTxveviTi sididea. maSin sworia, Tu ara, rom

)1,0(~/)( NSXn nn ? pasuxia, rom ara. am SemTxvevaSi samarTlia-

nia Semdegi

Teorema 8.1. Tu X1, X2,..., Xn damoukidebeli erTnairad normalu-

rad ganawilebuli SemTxveviTi sidideebia saSualoTi da dispersiiT2, maSin nn SXn /)( SemTxveviT sidides aqvs e.w. stiudentis anu

t -ganawileba Tavisuflebis xarisxiT n – 1 (an ufro mokled, t (n – 1)-ganawileba).

SevniSnoT, rom es ganawileba, damokidebulia parametrze, rome-

lsac saxelad Tavisuflebis xarisxi hqvia (da ara saSualo an dispe-

rsia, rogorc es normalurad ganawilebul populaciaSi iyo). es ter-

mini dakavSirebulia Semdeg mosazrebasTan: gavixsenoT 2nS SemTxveviTi

sididis ganmarteba: 2nS =

n

ini XX

n 1

2)(1

1. rogorc vxedavT, 2

nS war-

moadgens n cali SemTxveviTi sididis kvadratebis jams, magram TavisT-

avad es sidideebi, X1, X2,..., Xn SemTxveviTi sidideebis damoukideblobismiuxedavad, ar icvlebian damoukideblad, maT Soris arsebobs erTi

wrfivi kavSiri, kerZod,

0)(111111

n

ii

n

iin

n

ii

n

in

n

ii

n

ini XXXnXXXXX ,

92

rac imas niSnavs, rom Tu Cven viciT (X1- nX ), (X2- nX ), ..., (Xn- nX ) SemT-

xveviTi sidideebidan romelime n –1 cali, maSin gamovTvliT me-n-sac.amis gamo amboben, rom Tavisuflebis xarisxia n – 1.

davubrundeT 8.1 Teoremas da misi saSualebiT avagoT ndobis in-

tervali normaluri populaciis saSualosaTvis ucnobi dispersiis Se-mTxvevaSi. amisaTvis Cven ukve dagvWirdeba ara normaluri, aramed e. w.

t (n – 1)-ganawilebis kvantilebi, romlebsac Cven tn-1, p simboloTi aRvn-

iSnavT da ganvsazRvravT kvantilebis zogadi (6.13) gansazRvris safuZ-

velze Semdegnairad:

P{ t (n – 1) tn-1, p} = p. (8.6)

aqve gvinda aRvniSnoT, rom es ganawileba, iseve rogorc standa-

rtuli normaluri ganawileba simetrulia 0-is mimarT da normaluriganawilebis kvantilebis msgavsad (gaixseneT (6.14)), adgili aqvs tol-

obas:

tn-1, 1- p = - tn-1, p. (8.7)

amitomacaa, rom am ganawilebis kvantilebis cxrilebic mocemulia,

mxolod p 0.5-saTvis.sabolood, 8.2.1 SemTxvevis msgavsad, 8.1 Teoremaze dayrdnobiT,

t(n – 1)-ganawilebis kvantilebis cxrilebidan Cven SegviZlia yoveli pndobis albaTobis Sesabamisi tn-1, p kvantilis moZebna, aseve iseTi xp

ricxvis moZebna, rom

pxSXnP pnn /)( , anu

pnSxXnSxXP npnnpn // .

Sesabamisad, p ndobis albaTobis mqone anu 100 p%-ian ndobis

intervals ucnobi saSualosaTvis eqneba Semdegi saxe:

nsxxnsxx npnnpn // . (8.8)

wina SemTxvevis msgavsad, kavSiri tn-1, p da xp sidideebs Soris

Semdegnairia:

xp = tn-1, (1+p)/2. (8.9)

8.3. normaluri populaciis dispersiis intervaluri Sefaseba.

rogorc zemoT davinaxeT, populaciis dispersiis gadauadgileb-

ad wertilovan Sefasebas warmoadgens SerCeviTi dispersia. magram ro-

gor avagoT populaciis dispersiis intervaluri Sefaseba? amisaTvis

dagvWirdeba 2nS =

n

ini XX

n 1

2)(1

1SemTxveviTi sididis ufro sruly-

ofili daxasiaTeba, vidre mxolod misi maTematikuri lodinis codnaa.

es xerxdeba normalurad ganawilebuli populaciisaTvis (ufro zust-

93

ad, normaluri X1, X2,..., Xn SemTxveviTi sidideebisaTvis). kerZod, Sem-oviRoT Semdegi

gansazRvra 8.1. Tu X1, X2,..., Xn ~ N(0,1) da isini damoukidebelia,

maSin

n

iiX

1

2 SemTxveviTi sidide ganawilebulia e.w. 2 ganawilebis kan-

oniT, Tavisuflebis xarisxiT n (anu gvaqvs e. w. 2(n)-ganawileba).

normaluri da t()-ganawilebisagan gansxvavebiT 2()-ganawileba

(an rac igivea, 2() SemTxveviTi sidide) dadebiT naxevarRerZzea Tavm-oyrili da is asimetrulia. misi ricxviTi maxasiaTeblebia: E2() = ,D2() = 2, xolo roca > 2, misi moda (anu simkvrivis maqsimumiswertili) tolia m0 = - 2. 2()-ganawilebis simkvrives aqvs Semdegigrafiki:

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 2 3 4 5 6 7 8 9 10

mivubrundeT ndobis intervalis agebis amocanas normalurad ga-

nawilebuli populaciis dispersiisaTvis. davuSvaT, X1, X2, ..., Xn ~N(,2), maSin rogorc viciT, (Xi - )/, i = 1, 2, …, n SemTxveviTi sidid-

eebi ganawilebulia standartulad normalurad. amitom 8.1 gansazRvrisTanaxmad:

)1(~)(

),(~)( 2

12

22

12

2

nXX

nX n

i

nin

i

i

xolo .

aqedan gamomdinare davaskvniT, rom

)1(~)1( 2

2

2

nSn n

. (8.10)

dasaxelebuli -saTvis 2 ganawilebis cxrilebidan vipovoT2

2/1,1 n da 22/,1 n kvantilebi. maSin 1- ndobis albaTobis mqone

anu (1- )100%-ian ndobis intervals eqneba Semdegi saxe:

94

[ 2 2 2 21,1 / 2 1, / 2( 1) / , ( 1) /n n n nn S n S ]. (8.11)

8.4. binomuri populaciis p parametris intervaluri Sefaseba.

gavixsenoT, rom binomuri populaciis p parametris wertilovan

Sefasebas hqonda Semdegi saxe:n

XX

np

n

iin

1

1~ , np~ warmoidgineba

rogorc damoukidebeli da erTnairad ganawilebuli SemTxveviTi sidi-

deebis jami. amitom centraluri zRvariTi Teoremis Tanaxmad np~ asim-

ptoturad normaluradaa ganawilebuli parametrebiT, p da p(1-p)/n, sa-idanac Cven SegviZlia avagoT 1- ndobis albaTobis mqone ndobis int-

ervali, romelic unda amoixsnas Semdegi utolobidan:

nppzppnppzp nn /)1(~/)1(~2/12/1 .

cxadia, araviTar siZneles ar warmoadgens p-s mimarT kvadratu-li utolobis amoxsna, magram simartivis mizniT p(1-p)/n sidides uto-

lobis orive mxareSi cvlian misi SefasebiT npp nn /)~1(~ da sabolo-

od, 1- ndobis albaTobis mqone anu (1- )100%-ian ndobis intervalseqneba Semdegi saxe:

]/)~1(~~;/)~1(~~[ 2/12/1 nppzpnppzp nnnnnn . (8.12)

gavixsenoT 7.7 magaliTi, sadac np~ 0.0056 standartuli Secd-

oma toli iyo 0011.05000/9944.00056.0 . normaluri standartuli

ganawilebis kvantilebis cxrilebidan = 0.05-saTvis, vpoulobT, rom

z0.975 = 1.96, amitom 1- = 0.95 ndobis albaTobis mqone ndobis interalssabolood eqneba Semdegi saxe:

[0.0056-1.960.0011, 0.0056+1.960.0011] = [0.003444, 0.007756].gansxvavebiT am saxis ndobis intervalisa, romelic normalur

aproqsimaciazea damyarebuli, iyeneben ndobis intervalis agebis e.w.

zust meTodsac, romlis mixedviTac, ndobis intervals p albaTobisa-

Tvis uwodeben iseT [p1, p2] intervals, romlis p1 da p2 sazRvrebic

akmayofileben Semdeg Tanafardobebs:

n

xk

knkkn ppCppxXP )1(2/}|{ 111 ;

x

k

knkkn ppCppxXP

0222 )1(2/}|{ , (8.13)

sadac x aris binomuri SemTxveviTi sididis dakvirvebuli mniSvneloba.

8.5. puasonis ganawilebis parametris intervaluri Sefaseba.

95

puasonis ganawilebis parametris 1- ndobis albaTobis mqone

anu (1- )100%-iani ndobis interali moicema [1 / t, 2 / t] saxiT, sadac1 da 2 akmayofilebs gantolebebs (SeadareT (8.13)-s):

1

0

111

11

!1

!2/}|{

x

k

k

xk

k

ek

ek

xXP ,

1

0

212

2

!2/}|{

x

k

k

ek

xXP . (8.14)

aq x aris xdomilobaTa dakvirvebuli raodenoba da t aris xdomiloba-

weliwadSi.

rogorc gvaxsovs, 7.8 magaliTSi x=12, t=120000. amitom puason-is ganawilebis cxrilebidan vpoulobT 0.95 ndobis albaTobis mqone

ndobis Semdeg intervals -saTvis: [6.20, 20.96] da maSasadame, -saTvis:[6.20/120000, 20.96/120000] = [0.000052, 0.000175].

SevniSnoT, rom zogjer intervaluri Sefasebebis agebis amocana-

Si (iqneba es normaluri populaciis parametrebis, Tu binomuri da

puasonis ganawilebis parametrebis SemTxveva), saWiroa ara ormxrivi,

aramed calmxrivi (zeda an qveda) intervalebis ageba. aseT SemTxveveb-

Si (1-) albaTobis ndobis intervalis asagebad, saWiroa ganawilebaTa

kvantilebis cxrilebSi movZebnoT ara /2 da 1 - /2 donis, aramed

Sesabamisad da 1 - donis kvantilebi.

magaliTad, binomuri ganawilebis zeda da qveda calmxriv ndob-is intervalebs aqvT Semdegi saxeebi:

nppzpp nnn /)~1(~~1 (8.15)

da

nppzppnppzpp nnnnnn /)~1(~~/)~1(~~1 an . (8.16)

analogiurad iqneba sxva ganawilebebis SemTxvevaSi.

amocanebi1. cxrilSi mocemulia LVEF-ze (left ventricular ejection fraction)

dakvirvebis Sedegebi 27 pacientze, romlebsac aqvT kardiomiofatis

(cardiomyopathy) mwvave gafarToeba:pacienti LVEF pacienti LVEF

1 0.19 15 0.242 0.24 16 0.183 0.17 17 0.224 0.40 18 0.235 0.40 19 0.146 0.23 20 0.147 0.20 21 0.308 0.20 22 0.07

96

9 0.30 23 0.1210 0.19 24 0.1311 0.24 25 0.1712 0.32 26 0.2413 0.32 27 0.1914 0.28

1.1 gamoTvaleT LVEF-is saSualo;

1.2. gamoTvaleT LVEF-is standartuli gadaxra;1.3. gamoTvaleT LVEF-is saSualos standartuli Secdoma.

2. imisaTvis, rom daedginaT wnevianoba 5-6 wlis bavSvebSi, erT-

erT dabaSi 30 bavSvs gauzomes sisxlis diastoluri wneva da aRmoCn-

da, rom saSualo wnevis sidide Seadgens 56.2 mm vwy.sv.-s, xolo stan-

dartuli gadaxra ki 7.9 mm vwy.sv.-s. saerTo erovnuli gamokvlevidan

ki, cnobilia, rom am asakis bavSvebSi saSualo diastoluri wnevis si-

dide Seadgens 64.2 mm vwy.sv.-s.2.1. gvaqvs Tu ara safuZveli vamtkicoT, rom am dabaSi bavSvTa saSua-

lo diastoluri wneva gansxvavdeba saerTo-erovnuli gamokvlevidan

miRebuli saSualo diastoluri wnevis sididisagan?2.2. ageT 95%-iani ndobis intervali diastoluri sisxlis wnevis

standartuli gadaxrisaTvis aRniSnul 30 monacemze dayrdnobiT.3. erT-erTi hipoTezis mixedviT, glaukomaTi daavadebul adamia-

nebs aqvs saSualoze ufro maRali sisxlis wneva. amis dasadgenad 200

glaukomaTi daavadebuls gauzomes sisxlis wneva da aRmoCnda, rom sa-

Sualo wnevis sidide Seadgens 140 mm vwy.sv.-s, standaruli gadaxra ki

25 mm vwy.sv.-s.3.1. aageT 95%-iani ndobis intervali glaukomaTi daavadebuli popul-

aciis saSualo sisxlis wnevis WeSmariti mniSvnelobisaTvis;

3.2. Tu cnobilia, rom imave asakis janmrTel adamianebs aqvT saSualo

sistoluri wneva 140 mm vwy.sv., maSin aris Tu ara kavSiri glaukomasada sisxlis sistolur wnevas Soris?

4. davuSvaT xdeba gonoreis axali preparatis gamocda. 46 paci-

ents misces wamlis dRiuri doza 4grami. erTi kviris Semdeg aRmoCn-

da, rom 6 avadmyofi kvlav daavadebulia. aRvniSnoT p-Ti wamlis uefe-

qtobis albaToba.

4.1. rogoria p-s saukeTeso wertilovani Sefaseba?

4.2. rogoria 95%-iani ndobis intervali p-saTvis?4.3. davuSvaT, cvenTvis cnobilia, rom penicilin G-s 4.8 mega-erTeuli

dRiuri dozisaTvis uefeqtobis albTobaa 0.1. ra SegviZlia vTqvaT,

ori wamlis Sedarebaze?5. mwevelebis populaciaSi yoveldRiurad moweuli sigaretis

raodenobis Sesafaseblad SemTxveviT SearCies 16 mweveli. SerCevis sa-

97

Sualo aRmoCnda 14 sigareti dReSi, standartuli gadaxra ki 3.1 siga-reti dReSi.

5.1 gansazRvreT saSualos wertilovani Sefaseba;

5.2 ipoveT saSualos 95%-iani ndobis intervali.

6. 1996 wels centralur da samxreT amerikis 18 qveyanaSi, gam-

oiangariSes Sobadobis done yovel 1000 adamianze.

38 34 39 38 32 23 28 19 27

39 29 25 37 33 31 34 24 30

medianis da saSualos erTmaneTTan SedarebiT, ra daskvnis gakeTeba Se-

iZleba?

7. cnobilia, rom garkveuli medikamentis gamoyenebis Sedegadpulsis ricxvi matulobs. cnobilia, rom pulsis ricxvis standartu-

li gadaxra aris 5 dartyma wuTSi. SerCeul iqna am medikamentis 30

momxmarebeli da maTTvis pulsis ricxvis saSualo aRmoCnda 104 dar-

tyma wuTSi. vipovoT 99%-iani ndobis intervali pulsis ricxvis WeS-

mariti saSualosaTvis. igulisxmeba, rom Sesabamisi ganawileba daaxl-

oebiT normaluria.

8. centraluri respublikuri saavadmyofos 84 sxvadasxva adgi-

las gazomili xmauris donis saSualo iyo 61.2 decibali, xolo Sesw-

orebuli standartuli gadaxra 7.9 decibali. aageT 95%-iani ndobis

intervali saavadmyofoSi xmauris donis realuri saSualosaTvis.9. samSobiaro saxlis personals surs Seafasos axalSobilTa

wona. ra moculobis SerCeva iqneba saWiro, rom 90%-iani saimedoobiT

wonis realuri saSualo moTavsebuli iyos SerCevis saSualosagan 6

unciis (1 uncia = 28.3 gr) farglebSi, Tu cnobilia, rom axalSobil-

Ta wonis standartuli gadaxraa 8 uncia.

10. SemTxveviT SerCeuli 20 pacientisis 100 mililitr sisxl-

Si hemoglobinis saSualo Semcvleloba aRmoCnda 16 grami, xolo Ses-

worebuli standartuli gadaxra ki 2 gr. aageT 99%-iani ndobis int-

ervali WeSmariti saSualosaTvis.

11. meteorologma 15 civi haeris masivze dakvirvebis Sedegaddaadgina, rom maTi gavrcelebis saSualo siCqarea 18 mili/sT, xolo

standartuli gadaxra ki 2 mili/sT. aageT 95%-iani ndobis intervali

realuri saSualo siCqarisaTvis.

12. SemTxveviT arCeuli 6 spilos saSualo wonaa 12200 funti

(1 funti = 453.6 gr), xolo Sesworebuli standartuli gadaxra 200

funti. aageT 95%-iani ndobis intervali WeSmariti saSualosaTvis.

13. stresul situaciaSi myofi 6 qali guliscemis saSualo ri-

cxvi wuTSi Seadgens 115-s, xolo Sesworebuli standartuli gadaxra

aris 6. aageT 95%-iani ndobis intervali stresul situaciaSi myofi

qalebis guliscemis realuri saSualosaTvis.

98

14. moxucebis 500 momvlelisagan Semdgar SerCevaSi 60 mamaka-cia. ipoveT 90%-iani saimedoobis ndobis intervali moxucebis movlis

programaSi monawile mamakacTa WeSmariti proporciisaTvis.

15. dedaqalaqis 100 SemTxveviT SerCeul mcxovrebs Soris 27

msuqania. aageT 90%-iani ndobis intervali dedaqalaqis mosaxleobaSi

msuqani adamianebis realuri proporciisaTvis.

16. jandacvis saministros monacemebiT monacemebiT 13-14 wlis

mozardebidan yoveli mexuTe zogjer eweva sigarets. mTeli qveynis ma-

sStabiT 13-14 wlis mozardebSi sigaretis mwevelebis proporciasTan

Sesadareblad ganaTlebis saministrom gamokiTxa 200 13-14 wlis sko-

lis moswavle da daadgina, rom maTi 23% zogjer eweva sigarets.aageT 99%-iani ndobis intervali skolis 13-14 wlis moswavleebSi

sigaretis mwevelTa realuri proporcisasTvis da SeadareT is janda-

cvis saministros monacemebs.

17. garkveuli asakobrivi jgufis 100 gardacvlili adamianis

kvlevam aCvena, rom maTgan 25% gardaicvala kiboTi. aageT 98%-iani

ndobis intervali am asakobrivi jgufis gardacvlil adamianebSi im

adamianebis realuri proporciisaTvis, romlebic daiRupnen kiboTi.

18. medikoss surs 99%-iani saimedoobiTa da 2%-is sizustiT

Seafasos im qalebis realuri proporcia, romlebic Rebuloben vitam-

ins. adre Catarebuli kvlevis Tanaxmad 180 qalidan 25% Rebulobdavitamins. a). ra moculobis unda iyos saWiro SerCeva? b). Tu ar iqne-

boda xelmisawvdomi SerCeviTi proporcia, maSin ra moculobis SerCev-

is aReba mogviwevda?

19. mkvlevars surs 90%-iani saimedoobiTa da 5%-is sizustiT

Seafasos im mamakacebis reluri proporcia, romelTa simaRle 5 fut-

sa (1 futi = 30.48 sm) da 5 diumze (1 diumi = 2.54 sm) naklebia. a).

ra moculobis SerCevaa amisaTvis aucilebeli, Tu cnobilia, rom 300

mamakacidan 30-is simaRle 5 futsa da 5 diumze naklebia; b). ra mocu-

lobis SerCeva unda aviRoT im SemTxvevaSi, roca ar iqneba xelmisawv-

domi SerCeviTi proporcia?20. ipoveT TambaqoSi nikotinis Semcvlelobis dispersiisa da

standartuli gadaxrisaTvis 95%-iani ndobis intervali, Tu cnobi-

lia, rom 20 sigaretisagan Sedgenil SerCevaSi nikotinis Semcvlel-

obis Sesworebuli standartuli gadaxra 1.6 miligramis tolia.

21. 28 forToxlis Sesworebuli standartuli gadaxraa 0.34

sm. aageT 99%-iani ndobis intervali forToxlis diametris WeSmari-

ti standartuli gadaxrisaTvis.

22. 22 balaxis sakreWi manqanis mier 1 litri benziniT gakreW-

ili balaxis zolis sigrZis Sesworebuli SerCeviTi dispersiaa 2.6.

aageT 90%-iani ndobis intervali realuri dispersiisaTvis.

99

l e q c i a 8.

Tavi 9. hipoTezaTa Semowmeba. erTamokrefiani amocanebi.

9.1. Sesavali, ZiriTadi cnebebi.

wina leqciebSi (6 da 7) laparaki iyo populaciis ganawilebisucnobi parametrebis wertilovan da intervalur Sefasebebze. magram

xSirad mkvlevari winaswar gamoTqvams azrs ganawilebis parametris

Sesaxeb da Semdeg amowmebs, eTanxmeba Tu ara SerCeva parametrebis Ses-

axeb gamoTqmul ama Tu im winadadebas, romelsac mokled statistikurhipoTezas uwodeben. statistikur hipoTezebi SeiZleba daiyos or Zir-

iTad tipad: e.w. parametruli da araparametruli hipoTezebi. pirvelSemTxvevaSi hipoTeza gamoiTqmis populaciis ganawilebis parametrebis

Sesaxeb, xolo meore SemTxvevaSi, hipoTeza gamoiTqmis TviTon popul-

aciis ganawilebis an misi raime niSnis Sesaxeb. daviwyoT parametruli

hipoTezebis SeswavliT.magaliTi 9.1. davuSvaT, Seiswavleba qolesterinis Semcveloba

im adamianebis memkvidreebSi, romlebic erTi wlis win gardaicvalnen

gulis daavadebiT. ganixileba ori hipoTeza:

1) qolesterinis saSualo done memkvidreSi 175 mg/dl-ia;

2) qolesterinis saSualo done memkvidreSi 175 mg/dl-ze metia;

rogorc vxedavT, aq ori gamonaTqvamia populaciis (guliT gar-

dacvlilTa memkvidreebSi qolesterinis done) saSualos Sesaxeb, ro-

mlidanac pirvels nulovan hipoTezas, anu ZiriTad (Sesamowmebel) hi-

poTezas uwodeben, xolo meores ki mis alternatiul (anu garkveuli

azriT sawinaaRmdego) hipoTezas, an mokled, alternativas uwodeben.nulovan hipoTezas H0-iT aRniSnaven, xolo alternativas H1-iT. rat-

omaa ase mniSvnelovani hipoTezaTa Semowmebis (garCevis) amocana? Tun-

dac imitom, rom es aris swori gadawyvetilebis miRebis obieqturi

gza, subieqturi “marCielobis” sawinaaRmdegod, radgan es gza arsebiT-

ad eyrdnoba albaTobis Teoriis faqtebsa da maTematikurad mkacrad

dasabuTebul Teoremebs.

magaliTi 9.2. davuSvaT, Cven gvinda SeviswavloT moqmedebs Tu

ara dedis socialur-ekonomikuri mdgomareoba axalSobilis wonaze da

am mizniT davakvirdiT dabali socialur-ekonomikuri mdgomareobis

mqone 100 dedis pirmSos wonebs, saidanac gairkva, rom axalSobilTasaSualo wona aris 115 uncia, xolo standartuli gadaxra ki 24 un-cia, anu 115100 x da 24100 s . davuSvaT, CvenTvis cnobilia, rom mTe-

li populaciis masStabiT (magaliTad, milioni dakvirvebiT) axalSobi-

lTa saSualo wonaa = 120 uncia da = 25 uncia. gvaqvs Tu ara am

monacemebis mixedviT safuZveli vamtkicoT, rom SerCevidan miRebuli

100

saSualo wonis maCvenebeli dabalia populaciis Sesabamis maCvenebelze,anu dedis dabali socialur-ekonomikuri statusi uaryofiTad moqmed-

ebs axalSobilis wonaze?

am amocanis gadawyveta ucnobi saSualos intervaluri Sefasebis

meTodiT Cven ukve SegviZlia. amisaTvis davuSvaT, rom axalSobilTa

wonebi normaluradaa ganawilebuli ucnobi saSualoTi da standartu-

li gadaxriT = 25 uncia. SerCeviT monacemebze dayrdnobiT, avagoT

95%-iani ndobis qveda intervali saSualosaTvis, anu < c saxis int-

ervali. aseTi ndobis intervals asagebad saWiroa 1 - /2 donis kvant-ili ormxrivi intervalisaTvis Seicvalos 1 - donis kvantiliT. maS-

asadame, ndobis intervals eqneba saxe:

nzxn /1 ,

sadac = 0.05 da z1- = z0.95 =1.645. Sesabamisad, miviRebT intervals:

< 115 + 1.64525/10, anu < 119.1125 .

rogorc vxedavT, es intervali ar Seicavs = 120-s da Sesabam-isad, SerCeviT agebuli ndobis intervali marTlac sandoa ufro pat-

ara -saTvis da sabolood, vakeTebT daskvnas, rom dedis dabali so-

cialur-ekonomikuri statusi uaryofiTad moqmedebs axalSobilis wo-

naze.

magram am nawilSi Cveni mizania SevxedoT am da am tipis amocan-ebs hipoTezaTa Semowmebis amocanis kuTxidan. amocana ase yalibdeba:

davuSvaT, rom normalurad ganawilebuli populaciisaTvis, ro-

mlis dispersia cnobilia = 25, gamoTqmulia Semdegi nulovani da

alternatiuli hipoTeza:

H0 : = 0,H1 : < 0.

Cven vuSvebT, rom samarTliani SeiZleba iyos mxolod erTi am

ori hipoTezidan, magram radgan nulovani hipoTeza ufro mniSvnelova-

nia CvenTvis, terminologiac maszea agebuli. kerZod, imis magivrad,rom vTqvaT, “monacemebis mixedviT H0 hipoTezaa samarTliani”, Cven

vambobT: “monacemebis mixedviT H0 hipoTezis uaryofis safuZveli ara

gvaqvs”. piriqiT, Tu aRmoCndeba, rom “monacemebis mixedviT H1 hipoTe-

zis uaryofis safuZveli ara gvaqvs”, maSin Cven calsaxad uarvyofT

H0 hipoTezas. sakiTxis ase dayenebisas, hipoTezaTa garCevis procesSi

Cven SeiZleba ori tipis (gvaris) Secdoma davuSvaT da orive tipis

Secdomas Tavisi albaToba gaaCnia:

I gvaris Secdomis albaToba, es imis albaTobaa, rom uarvyoT

sinamdvileSi samarTliani (swori) H0 hipoTeza.

II gvaris Secdomis albaToba, es imis albaTobaa, rom ar uarvy-oT sinamdvileSi arasamarTliani (araswori) H0 hipoTeza.

101

Cveni magaliTisaTvis, I gvaris Secdomis albaToba, iqneboda al-baToba imisa, rom gveTqva axalSobilis wona naklebia 120 unciaze, ma-Sin, roca is faqtiurad 120 unciis tolia, xolo II gvaris SecdomisalbaToba, iqneboda albaToba imisa, rom gveTqva axalSobilis wona

tolia 120 unciisa, maSin, roca is faqtiurad 120 unciaze naklebia.

ras niSnavs yovelive es? magaliTad, mogvivida I gvaris Secdoma, maSinTiTqosda saWiro iqneboda damatebiTi saxsrebi bavSvebis gamojanmrTe-

lebisaTvis, magaram sinamdvileSi es saWiro ar aris. piriqiT, vTqvaT,

mogvivida II gvaris Secdoma, maSin TiTqosda yvelaferi karagadaa, mag-

ram am dros marTlac saWiroa damatebiTi xarjebis gaReba bavSvebis

janmrTelobis gasaumjobeseblad. ase, rom Cven vxedavT, rom I da IIgvaris Secdomebs da maT albaTobebs pirdapiri kavSiri aqvT fulTan.

I gvaris Secdomis albaToba -Ti aRiniSneba da mas kriteriumismniSvnelovnebis dones uwodeben.

II gvaris Secdomis albaToba -Ti aRiniSneba da 1 - sidides

kriteriumis simZlavre hqvia. SevniSnoT, rom kriteriumis simZlavreimis albaTobaa, rom uarvyoT sinamdvileSi araswori H0 hipoTeza.

am leqciis danarCen nawilebSi Cven gadavalT uSualod hipoTe-

zaTa Semowmebis amocanebze, populaciebis sxvadasxva ganawilebebis

dros.

9.2. hipoTezis Semowmeba normaluri populaciis saSualosaTvis

cnobili dispersiis dros.

am amocanis erTi nawili Cven faqtiurad gavarCieT 9.2 magaliT-

is saxiT, Tumca ndobis intervalebis terminebSi da daaxloebiT Seviq-

meniT warmodgena imaze, Tu rogor unda CamovayaliboT igive saxis am-

ocanebi hipoTezebis terminebSi. amitom Cven ar gavacalkevebT aseTi

sami tipis amocanas da SevecdebiT paralelurad amovxsnaT isini maTi

msgavsebis gamo.

normalurad ganawilebuli populaciisaTvis, romlis dispersiisricxviTi mniSvneloba cnobilia, arsebobs nulovani hipoTezisa da al-

ternativis Camoyalibebis sami SesaZlebloba:

1) H0 : = 0

H1 : < 0

2) H0 : = 0

H1 : > 0

3) H0 : = 0

H1 : 0

1) da 2) SemTxvevaSi amboben, rom saqme aqvT calmxriv alterna-tivebTan, xolo 3)-Si ki ormxriv alternativasTan.

amocanis gadasaWrelad, samive SemTxvevaSi asaxeleben I gvarisSecdomis albaTobas. es is albaTobaa rogor Secdomazec Tanaxma

unda viyoT samarTliani hipoTezis uaryofis SemTxvevaSi. radgan saqme

SemTxveviTobas exeba, ar SeiZleba, rom iyos 0-is toli. gavixsenoT,

102

Tundac 9.2 magaliTi da davuSvaT, rom = 0. maSin normaluri ganaw-

ilebis kvantilebis cxrilebidan z1- = z1 = da nzxn /1 =

= , anu miviRebdiT, rom < , rac arafris mTqmelia -s Sesaxeb.magram miuxedavad amisa, ra Tqma unda, gvinda rom iyos SesaZleblad

mcire. ase, rom asaxeleben raime mcire -s, ise rom > 0, magaliTad,

= 0.05, =0.01.samive SemTxvevaSi saWiroa gadawyvetilebis miRebis wesis anu

e.w. statistikuri kriteriumis ageba: rodis uarvyoT da rodis aranulovani hipoTeza. statistikuri kriteriumis ageba sazogadod, arse-

biTadaa damokidebuli imaze, Tu ris mimarTaa gamoTqmuli nulovani

hipoTeza (Cvens SemTxvevaSi esaa normaluri ganawilebis saSualo )da gulisxmobs iseTi kritikuli (sakritiko) aris agebas, romelSic

SerCveiTi saSualos (radgan nulovani hipoTeza gamoTqmulia popula-

ciis saSualos Sesaxeb da misi saukeTseo wertilovani Sefasebaswored SerCeviTi saSualoa) moxvedra nulovani hipoTezis gakritik-

ebas, anu uaryofas niSnavs. maSasadame, mTavari amocanaa kritikuli ar-

is ageba. rogori unda iyos kritikuli are? cxadia, rom kritikuli

aris albaToba hipoTezis samarTlianobis SemTxvevaSi unda emTxveodes

(yovel SemTxvevaSi, ar aRematebodes) dasaxelebul mcire -s da swo-

red es udevs safuZvlad kritikuli aris agebas. Sinaarsobrivad gasag-ebia, rom Tu gansxvaveba wertilovan Sefasebasa da Sesafasebel popu-

laciis parametrs Soris sagrZnobia, maSin albaT nulovani hipoTeza

ar unda iyos swori. amitom saSualos Sesaxeb hipoTezis Semowmebis

amocanaSi, azriania vilaparakoT 0nx sxvaobis ricxviT mniSvnelo-

baze da Sesabamisad, 0nX sxvaobaze, rogorc SemTxveviT sidideze.

gasagebia, rom Tu nx “dasaSvebze” mcirea, maSin 1) amocanaSi nulovani

hipoTeza unda uarvyoT; aseve Tu nx “dasaSvebze” didia, maSin ukve 2)

amocanaSi unda uarvyoT nulovani hipoTeza. rac Seexeba 3) amocanas,

am SemTxvevaSi nulovan hipoTezas uarvyofT maSin, roca 0nx iqne-

ba “dasaSvebze” didi. aRvniSnoT es “dasaSvebi” (kritikuli) mniSvnel-

obebi Sesabamisad, C1,M C2-iTa da C3-iT. maSin, rogorc vTqviT, kriti-

kul areebs aqvT Sesabamisad, Semdegi saxeebi:

1) nx < C1, 2) nx > C2 da 3) 0nx > C3. (9.1)

maSasadame, Cveni amocana davida C1,M C2 da C3 ricxvebis povna-

ze. maT sapovnelad gavixsenoT, rom kritikuli aris albaToba hipoTe-

zis samarTlianobis SemTxvevaSi unda emTxveodes dasaxelebul -s.ase, rom saWiroa iseTi C1,M C2 da C3 ricxvebis povna, rom:

1) 01 |CXP n , 2) 02 |CXP n ,

103

3) 030 |CXP n . (9.2)

rogorc viciT, normaluri populaciis SemTxvevaSi, nulovani

hipoTezis samarTlianobis dros (e.i., roca =0), /)( 0 nXn

SemTxveviT sidides aqvs standartuli normaluri ganawileba. amitom

00 |/)( zXnP n ,

010 |/)( zXnP n ,

02/10 |/|| zXnP n ,

saidanac davaskvniT, rom:

1) C1 = nz /0 , 2) C2 = nz /10 = nz /0 ,

3) C3 = nz /2/1 .

sabolood, statistikuri kriteriumi (gadawyvetilebis miRebis

wesi) ase yalibdeba:

1) amocanaSi, Tu nx < nz /0 , maSin mniSvnelovnebis doniT

H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli ara

gvaqvs;

2) amocanaSi, Tu nx > nz /0 , maSin mniSvnelovnebis doniT

H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli ara

gvaqvs;

3) amocanaSi, Tu 0nx > nz /2/1 , maSin mniSvnelovnebis

doniT H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli

ara gvaqvs.

sailustraciod davubrundeT 9.2 magaliTs da vnaxoT rogor

muSaobs Cvens mier Camoyalibebuli statistikuri kriteriumi.

rogorc gvaxsovs, 115100 x , = 25 da saqme exeboda 1) tipis

amocanas: H0 : = 120, H1 : < 120.vTqvaT, = 0.05. maSin vinaidan z0.05 = -z0.95 = -1.645, Cveni krite-

riumidan nz /0 = 120 –1.64525/10 = 120 – 4.1125 = 115.8875.

radgan 115100 x <115.8875, amitom = 0.05 mniSvnelovnebis doniT H0

hipoTezas uarvyofT. rogorc vxedavT, pasuxi igive darCa, rac 9.2 ma-

galiTSi.

9.3. hipoTezis Semowmeba normaluri populaciis saSualosaTvis

ucnobi dispersiis dros.

amocanebis dasma xdeba zustad iseve, rogorc 9.2 punqtSi: Sesa-

mowmebelia nulovani hipoTeza H0 : = 0 sami sxvadasxva tipis al-

ternativisaTvis:

104

1) H1 : < 0 ; 2) H1 : > 0; 3) H1 : 0.kritikuli areebis dasadgenad, rogorc wina punqtSi vnaxeT,

arsebiTi iyo Tn /)( 0 nXn SemTxveviT sididis ganawileba,

romelsac mokled kriteriumis statistikas uwodeben. kerZod, cnobi-

li -s dros mas hqonda standartuli normaluri ganawileba. gavixse-

noT, rom ucnobi -s SemTxvevaSi ndobis intervalis agebis amocanaSi

n/ saSualos standartul Secdomas vcvlidiT nSn / sididiT.

moviqceT axlac ase da kriteriumis statistikad ganvixiloT SemTxve-

viTi sidide: Tn nn SXn /)( 0 . maSin Teorema 8.1-is Tanaxmad, am

SemTxveviT sidides aqvs t(n-1)-ganawileba. danarCeni msjeloba wina pu-

nqtSi moyvanili msjelobis analogiuria. gansxvaveba imaSia, rom norm-aluri standartuli ganawilebis kvantilebi Seicvleba t(n-1)-ganawil-

ebis Sesabamisi kvantilebiT da sabolood, statistikuri kriteriumi

(gadawyvetilebis miRebis wesi) ase Camoyalibdeba:

1) amocanaSi, Tu nx < nst nn /,10 , maSin mniSvnelovnebis don-

iT H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli ara

gvaqvs;

2) amocanaSi, Tu nx > nst nn /,10 , maSin mniSvnelovnebis don-

iT H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli ara

gvaqvs;

3) amocanaSi, Tu 0nx > nst nn /2/1,1 , maSin mniSvnelovnebis

doniT H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli

ara gvaqvs.

am kriteriumis gamoyenebis sailustraciod kvlav mivubrundeT

9.2 magaliTs, sadac 11561 x da 2461 s . vTqvaT, = 0.05. maSin rad-

gan t60, 0.05 = -t60, 0.95 = -1.671, Cveni kriteriumidan nst nn /,10 =120–1.67124/7.82 = 120 – 5.04 = 114.96. radgan 11561 x >114.96, = 0.05 mniS-

vnelovnebis doniT H0 hipoTezis uaryofis safuZveli ara gvaqvs.

9.4. kriteriumis simZlavris gamoTvla.

rogorc 9.2 da 9.3 punqtebidan davinaxeT, gansxvaveba maT Sor-

is mxolod kriteriumis statistikis ganawilebaSi iyo, danarCenimsjelobebi zustad emTxveoda erTmaneTs. igive SeiZleba iTqvas am pun-

qtSi Sesaswavl sakiTxzec. amitom Cven mxolod 9.2 punqtisaTvis gam-

oviTvliT kriteriumis simZlavreebs. rogorc gvaxsovs, kriteriumis

simZlavre ganisazRreba rogorc araswori nulovani hipoTezis uaryo-

fis albaToba. magram araswori hipoTeza, avtomaturad niSnavs alter-

nativis samarTlianobas, xolo nulovan hipoTezas uarvyofdiT maSin,

105

roca kriteriumis statistika Rebulobda garkveul mniSvnelobas kri-tikuli aridan. amitom kriteriumis simZlavre SeiZleba ganvsazRvr-

oTOrogorc kritikuli aris alternatiuli albaToba. amitom H0 : == 0 nulovan hipoTezasTan davazustoT alternativa Semdegnairad H1 : = 1 < 0 da gamovTvaloT kritikuli aris alternatiuli albaToba,

anu kritikuli aris albaToba, roca samarTliania alternativa = 1.

9.2 punqtidan gavixsenoT, rom calmxrivi marcxena kritikuli are ga-

nisazRreboda nx < nz /0 utolobiT, romelic miviReT ganto-

lebidan:

00 |/)( zXnP n .

amitom yovelive zemoTqmulis gamo kriteriumis simZlavre tolia

10 |/)(1 zXnP n . (9.3)

magram alternativis samarTlianobis dros /)( 0 nXn Se-

mTxveviT sidides ara aqvs standartuli normaluri ganawileba. am

dros standartulad normalurad ganawilebulia /)( 1 nXn Sem-

TxveviT sidide. amitom (9.3) gadavweroT ase:

1101 |/)(/)(1 nzXnP n , (9.4)

saidanac sabolood davaskvniT, rom calmxrivi marcxena kriteriumis

simZlavre = 1(<0) alternativis dros tolia:

/)(1 10 nz . (9.5)

analogiuri msjelobiT SegviZlia davaskvnaT, rom calmxrivi ma-

rjvena kriteriumis simZlavre = 1(>0) alternativis dros tolia:

/)(1 01 nz . (9.6)

dabolos, ormxrivi kriteriumis simZlavre = 1(0) alternativis

dros tolia:

/||1 012/ nz . (9.7)

Tu davakvirdebiT (9.5), (9.6) da (9.7) formulebs, aRmovaCenTSemdeg efeqtebs, romlebic moqmedebs kriteriumis simZlavreze:

a) Tu mcirdeba, maSin z ( / 2z ) mcirdeba da funqciis zrdadobis

gamo mcirdeba simZlavre;

b) SerCevis moculobis zrda iwvevs simZlavris gazrdas;

g) rac ufro didia gansxvaveba parametris hipoTetur da altrnati-

ul mniSvnelobebs Soris, miT ufro didia simZlavre;d) rac ufro didia standartuli gadaxra, miT ufro mcirea simZlav-

re.

TiToeuli es efeqti Cven SegviZlia davinaxoT magaliTze. sail-

ustraciod gavarCioT 9.1 magaliTiis Semdegi modifikacia:

106

magaliTi 9.3. vTqvaT, Seiswavleba qolesterinis Semcveloba imadamianebis memkvidreebSi, romlebic erTi wlis win gardaicvalnen gu-

lis daavadebiT. davuSvaT, cnobilia, rom qolesterinis done normal-

uradaa ganawilebuli da standartuli gadaxra 50-is tolia. 10 bavSv-

is monacemebis mixedviT, qolesterinis saSualo done aRmoCnda 200mg/dl. Cven gvainteresebs kriteriumis simZlavre nulovani hipoTezisa:

qolesterinis saSualo done memkvidreebSi 175 mg/dl-ia, alternativ-

is mimarT, rom qolesterinis saSualo done ufro maRalia da Seadg-

ens 190 mg/dl-s.

ganvixiloT ori SemTxveva, = 0.05, = 0.01. radgan saqme gvaqvsmarjvena calmxriv kriteriumTan, amitom simZlavris gamosaTvlelad

orive SemTxvevaSi viyenebT (9.6) formulas, romlis mixedviTac

/)(1 01 nz . roca =0.05, maSin z0.05= - z0.95= -1.645.

amitom gvaqvs

1 - = (-1.645 + 3.17(190-175)/50) = (-0.694) == 1- (0.694) 1-0.755 = 0.245.

roca = 0.01, z0.01 = - z0.99 = -2.326 da amitom

1 - = (-2.326 + 3.17(190-175)/50) = (-1.375) == 1- (1.375) 1-0.9158 = 0.0842.

rogorc vxedavT, kriteriumis simZlavrem marTlac daiklo.

xSirad, rodesac calkeuli eqsperimentis Catareba sakmaod Zvi-

ri jdeba, saWiroa eqsperimentis winaswar dagegmva da imis gansazRvraTu cdebis ra minimaluri raodenoba iqneba sakmarisi kriteriumis das-

axelebuli simZlavris misaRwevad mocemuli mniSnelovnebis donis

dros. (9.5), (9.6) da (9.7) formulebi iZleva amis saSualebas da amas

Cven momdevno punqtSi vnaxavT.

9.5. SerCevis minimaluri moculobis gansazRvra.

Cveni amocana am punqtSi ase yalibdeba: rogori unda iyos SerC-

evis moculoba, rom mocemuli mniSvnelovnebis donisaTvis kriteri-

umis simZlavre meti an toli iyos winaswar dasaxelebul 1- ricxv-

ze?

rogorc (9.5), (9.6) da (9.7) formulebidan gvaxsovs kriteriu-

mis simZlavreebi Sesabamisad tolia:

/)( 10 nz , (9.5’)

/)( 01 nz , (9.6’)

/|| 012/ nz . (9.7’)

axla Cveni amocanaa amovxsnaT Semdegi saxis utolobebi:

107

1/)( 10nz , 1/)( 01nz ,

1/|| 012/ nz .

cxadia, rom es utolobebi Sesabamisad eqvivalenturia Semdegi

utolobebisa:

110 /)( znz , 101 /)( znz ,

1012/ /|| znz ,

saidanac calmxrivi alternativebis dros miviRebT, rom:2

102

112 )/()( zzn , (9.8)

xolo ormxrivi alternativisaTvis ki gveqneba:2

102

12/12 )/()( zzn . (9.9)

sailustraciod ganvixiloT 9.3 magaliTis Semdegi modifikacia.magaliTi 9.4. vTqvaT, Seiswavleba qolesterinis Semcveloba im

adamianebis memkvidreebSi, romlebic erTi wlis win gardaicvalnen gu-

lis daavadebiT. davuSvaT, cnobilia, rom qolesterinis done normal-

uradaa ganawilebuli da standartuli gadaxra 50-is tolia. nulova-

ni hipoTezaa: qolesterinis saSualo done memkvidreebSi 175 mg/dl-ia,

alternativis mimarT, rom – qolesterinis saSualo done ufro maRa-

lia da Seadgens 190 mg/dl-s. Cveni amocanaa ganvsazRvroT, Tu SerCev-

is ra minimaluri moculoba iqneba sakmarisi 5%-iani mniSvnelovnebis

donis dros 90%-iani simZlavris misaRwevad.

amoxsna. radgan saqme gvaqvs marjvena calmxriv kriteriumTan,amitom viyenebT (9.8) formulas, saidanac

2

102

112 )/()( zzn

1.95225/)28.1645.1(2500)175190/()(50 2229.095.0

2 zz .

Sesabamisad, davaskvniT, rom saWiroa 96 dakvirveba.

9.6. hipoTezis Semowmeba normaluri populaciis

dispersiisaTvis (ormxrivi alternativa).

Cveni amocanaa SevamowmoT martivi hipoTeza normaluri popula-

ciis dispersiis Sesaxeb, anu

H0 : 2 = 20 hipoTeza H1 : 2 2

0 alternativis winaaRmdeg.

dispersiisaTvis ndobis intervali igeboda im faqtze dayrdnob-

iT, rom )1(~)1( 2

2

2

nSn n

, Tuki 2

nS warmoadgens SerCeviT dispersias

agebuls damoukidebeli normaluri SemTxveviTi sidideebiT, romelTa

dispersiaa 2 (me-7 leqcia). amitom kriteriumis statistikad aviRoT

swored 20

2 /)1( nSn SemTxveviTi sidide, romelsac hipoTezis samarT-

108

lianobis dros aqvs 2(n-1) ganawileba. statistikuri kriteriumi Sem-degnairad Camoyalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis kriteriumis

statistikis ricxviTi mniSvneloba akmayofilebs pirobas2

2/1,120

222/,1 /)1( nnn sn , (9.10)

maSin H0 hipoTezis uaryofis safuZveli ara gvaqvs. winaaRmdeg SemTxv-

evaSi mas uarvyofT.

9.7. hipoTezis Semowmeba binomuri populaciis p parametrisSesaxeb (ormxrivi alternativa).

Cveni amocanaa SevamowmoT martivi hipoTeza binomuri populaci-

is p parametris (proporciis) Sesaxeb, anuH0 : p = p0 hipoTeza H1 : p p0 alternativis winaaRmdeg.

me-7 leqciis mixedviT, np~ = X / n SemTxveviTi sidide, sadac X

aRniSnavs binomur SemTxveviT sidides parametriT p, asimptoturadnormaluradaa ganawilebuli parametrebiT: = p da 2 = p(1-p)/n. amit-

om kriteriumis statistikad viRebT)1(

~

00

0

pnp

ppT n

n

SemTxveviT sid-

ides, romelic hipoTezis samarTlianobis dros asimptoturad standa-

ruli normaluria da statistikuri kriteriumi zustad iseTive iqne-

ba, rogoric iyo saSualos Sesaxeb hipoTezis Semowmebis amocanaSi no-

rmaluri populaciisaTvis ormxrivi alternativis dros:

Tu dasaxelebuli mniSvnelovnebis donisaTvis Tn statistikisdakvirvebuli mniSvneloba akmayofilebs pirobas

2/1

00

02/

)1(

~

z

pnp

ppz n , (9.11)

maSin H0 hipoTezis uaryofis safuZveli ara gvaqvs. winaaRmdeg SemTxv-

evaSi mas uarvyofT.

gamovTvaloT am kriteriumis simZlavre H1 : p = p1 p0 alterna-

tivis dros. 9.4 punqtSi Catarebuli msjelobis msgavsad, SegviZlia

davaskvnaT, rom kriteriumis simZlavre am alternativisTvis iqneba:

npp

ppz

pp

pp

)1(

||

)1(

)1(1

00

102/

11

00 . 9.12)

9.5 punqtis analogiuri msjelobiT vpoulobT SerCevis minima-

luri moculobis formulas:2

00

112/12/12

10

00

)1(

)1(

)(

)1(

pp

ppzz

pp

ppn . (9.13)

109

aRniSnuli kriteriumi damyarebulia binomuri ganawilebis norm-

alur aproqsimaciaze. es ki maSinaa SesaZlebeli, roca np0(1- p0) 5.zogjer ki, saWiroa hipoTezis Semowmeba, roca es piroba darRveulia.

amitom iyeneben hipoTezis Semowmebis zust meTodsac (gavixsenoT ndo-bis intervalis agebis zusti meTodi binomuri ganawilebis parametri-

saTvis me-7 leqciidan).

hipoTezis Semowmebis amocanaSi, iTvlian e.w. p-mniSvnelobas,romelic ase ganimarteba:

Tu x/n p0, sadac x aris binomuri SemTxveviTi sididis dakvirv-ebuli mniSvneloba, maSin

x

k

knkkn ppCppxXPp

0000 )1(}|{2/ , 9.14)

xolo Tu x/n > p0, maSin

n

xk

knkkn ppCppxXPp )1(}|{2/ 000

. (9.15)

Tu p-s gamoTvlili mniSvneloba aRmoCndeba -ze naklebia, maS-

in amboben, rom Sedegi (laparakia x-ze) statistikurad mniSvnelovania,winaaRmdeg SemTxvevaSi – is statistikurad umniSvneloa da nulovan

hipoTezas ar uaryofen.

am kriteriumis mniSvnelovnebis sailustraciod moviyvanoT

magaliTi 9.5. birTvuli energiebis qarxanaSi momuSave 55-64ww.

asakis mamakacTa Soris gardacvlili 13 mamakacidan 5-is gardacvalebis

mizezi aRmoCnda kibo. cnobilia, rom mamakacTa am asakobrivi jgufisa-

Tvis gardacvalebaTa 20% modis kibos garkveul formaze. aris Tu

ara qarxanaSi momxdari es Sedegi mniSvnelovani (da maSasadame, birTv-

uli energiebis qarxanaSi muSaoba saSiSi)?

amoxsna. Cveni amocanaa SevamowmoT H0 : p = 0.2 hipoTeza, H1 : p 0.2 alternativis winaaRmdeg. gamovTvaloT np0(1-p0) = 130.20.8= 2.1. vi-naidan 2.1<5, amitom normalur aproqsimacias ver gamoviyenebT. gamovT-

valoT x/n = 5/13 =0.38 > p0 = 0.2. Sesabamisad, p-mniSvnelobis dasadgenad

viyenebT (9.15) formulas

4

0

1313

13

5

1313 8.02.0128.02.02

k

kkk

k

kkk CCp .

binomuri ganawilebis cxrilebidan n = 13 da p = 0.2-saTvis vpou-lobT: P{X =0}=0.055, P{X =1}=0.1787, P{X =2}=0.268, P{X = 3}=0.2457,P{X = 4}=0.1535, saidanac p = 2(1-(0.055+0.1787+0.268+0.2457+0.1535)) =0.198.

radgan p = 0.198 > 0.05 davaskvniT, rom x = 5 Sedegi statistiku-

rad umniSvneloa, anu birTvuli energiebis qarxanaSi muSaoba saSiSi

ar aris.

110

9.8. hipoTezis Semowmeba puasonis populaciis parametrisSesaxeb (mcire moculobis SerCevebisaTvis).

binomuri populaciis parametris Sesaxeb hipoTezis Semowmebismsgavsad, mcire moculobis SerCevis SemTxvevaSi, aqac viyenebT p-mni-Svnelobis meTods H0 : = 0 hipoTezis Sesamowmeblad H1 : 0

alternativis winaaRmdeg.

Tu x 0, sadac x aris binomuri SemTxveviTi sididis dakvirveb-uli mniSvneloba, maSin

x

k

k

ek

xXPp0

00

0

!}|{2/

, (9.16)

xolo Tu x > 0, maSin

1

0

00

0

!1}|{2/

x

k

k

ek

xXPp . (9.17)

Tu p-s gamoTvlili mniSvneloba aRmoCndeba -ze naklebi, maS-

in amboben, rom Sedegi (laparakia x-ze) statistikurad mniSvnelovania,winaaRmdeg SemTxvevaSi – is statistikurad umniSvneloa da nulovan

hipoTezas ar uaryofen.

didi moculobis SemTxvevaSi aqac iyeneben normalur aproqsima-

cias, kerZod, im faqts, rom (X - 0)2 /0 ~ N 2(0;1) ~ 2(1).

amocanebi1. SC-s (serum-creatinine) donis saSualo mniSvneloba 12 pacien-

tis mixedviT axali preparatis miRebidan 24 saaTis Semdeg aRmoCnda1.2 mg/dl.

1.1. davuSvaT, generaluri populacia ganawilebulia normalurad sa-

SualoTi 1 mg/dl da standartuli gadaxriT 0.4 mg/dl. = 0.05 mniS-

vnelovnebis doniT SeamowmeT hipoTeza: “SC-s saSualoa 1.2 mg/dl”

alternativis winaaRmdeg: “SC-s saSualo mniSvneloba araa 1 mg/dl-is

toli”;

1.2. ras udris kriteriumis p-mniSvneloba?

1.3. rogoria 1- ndobis albaTobis mqone ndobis intervali SC-s WeS-mariti saSualo mniSvnelobisaTvis?

1.4. davuSvaT, 1.1-Si ucnobia populaciis standartuli gadaxra, magram

mocemulia SerCeviTi standartuli gadaxra, romelic 0.6 mg/dl-is

tolia. SeamowmeT am SemTxvevaSi 1.1-Si Sesamowmebeli hipoTeza;

1.5. rogor Seicvleba pasuxebi 1.1 – 1.4 punqtebSi, Tuki alternativa

iqneba: “SC-s saSualo mniSvneloba metia 1 mg/dl-ze”?

2. yovelwliuri dakvirvebebis mixedviT, 0-4 wlis asakis bavSve-

bSi asTma aRiricxeba biWebis 1.4%-Si da gogonebis 1%-Si.

111

2.1. davuSvaT mocemuli weliwadis ganmavlobaSi 0-4 wlis asakis 500biWidan asTma aRmoaCnda 10 bavSvs, romelTa dedebic ewevian sigarets.

aris Tu ara mniSvnelovani gansxvaveba am jgufsa da generalur popu-

lacias Soris?

2.2. rogoria p-mniSvneloba 2.1 amocanaSi?

2.3. davuSvaT, mocemuli weliwadis ganmavlobaSi 0-4 wlis asakis 300gogonadan asTma aRmoaCnda 4 bavSvs, romelTa dedebic ewevian sigar-

ets. aris Tu ara mniSvnelovani gansxvaveba am jgufsa da generalur

populacias Soris?


3. dakvirvebebis mixedviT, fexmZime qalebis 30% uCivis nausea-sfexmZimobis 24-28 kviris ganmavlobaSi. gamokiTxes 200 qali, romelic

am xnis ganmavlobaSi regularulad Rebulobda eritromicins da aRmo-

Cnda, rom 110 maTgani uCivis nausea-s. SeamowmeT hipoTeza imis Sesax-eb, rom nausea-Ti daavadebis done am jgufsa da generalur populaci-

aSi erTnairia.

4. rkinis-deficitis anemia kvebis higienis mniSvnelovani probl-

ema iyo aSS-Si. 1951 wels Catarda SedarebiT dabali Semosavlebis

mqone ojaxebSi 9-11 wlis biWebis gamokvleva da aRmoCnda, rom rkinis

miRebis dRiurma saSualo sididem am jgufSi Seadgina 12.50 mg, stand-

artuli gadaxriT 4.75 mg. davuSvaT, rom am asakisaTvis mTliani popu-laciis rkinis miRebis saSualo sidide Seadgens 14.44mg-s. Cven gvaint-

eresebs SevamowmoT hipoTeza, rom SedarebiT dabali Semosavlebis mqo-

ne ojaxebSi 9-11 wlis biWebis rkinis miRebis sidide gansxvavebulia

generaluri populaciis imave sididisagan.

4.1. CamoayalibeT Sesabamis nulovani da alternatiuli hipoTezebi;

4.2. = 0.05 mniSvnelovnebis doniT SeamowmeT nulovani hipoTezis sa-

marTlianobis sakiTxi;

4.3. rogoria p-mniSvneloba 4.2 amocanaSi?davuSvaT, generaluri populaciis dRiuri rkinis miRebis stan-

dartuli gadaxra tolia 5.56mg-isa. Cven gvinda SevamowmoT aris Tu

ara dabali Semosavlebis mqone ojaxebSi 9-11 wlis biWebis rkinis mi-

Rebis dRiuri normis standartuli gadaxra generaluri populaciis

imave maCvenebelTan Sesadari.

4.4. CamoayalibeT Sesabamis nulovani da alternatiuli hipoTezebi;

4.5. = 0.05 mniSvnelovnebis doniT SeamowmeT nulovani hipoTezis sa-marTlianobis sakiTxi;


4.7. rogoria 1- ndobis albaTobis mqone ndobis intervali dabali

Semosavlebis mqone ojaxebSi 9-11 wlis biWebis rkinis miRebis dRiuri

normis standartuli gadaxrisaTvis?

112

5. 64 cxvars gaukeTes ineqcia 10mg/ml doziT. erTi saaTis Sem-deg antibiotikis koncentracia saSualod iyo 24.857mg/ml, standart-

uli gadaxriT 3.924mg/ml. ipoveT 95%-iani ndobis intervali popul-

aciis saSualo koncentraciisaTvis.

6. sxvadasxva adgilas napovni niangis kvercxebis saSualo rao-

denobaa 15, xolo standartuli gadaxra ki 7. aageT 90%-iani ndobis

intervali niangis populaciis kvercxebis saSualosaTvis.

7. baltiis zRvaSi, gdanskTan, wylis 8 sinjSi stronciumis ko-

ncentracia iyo: 4.0 5.5 4.5 6.0 5.75 4.5 4.0. daaxasiaTeT

gdanskTan zRvis wylis mdgomareoba stronciumTan mimarTebaSi.

8. 2003 wels axal zelandiaSi dabadebuli 69224 bavSvidan33937 gogo iyo.

8.1. SeafaseT gogonebis dabadebis albaToba;

8.2. aageT 95%-iani ndobis intervali;

8.3. aris Tu ara gogos da biWis dabadeba tolalbaTuri?

9. Sokoladis mwarmoebeli, yoveli sawarmoo ciklis dasrule-

bis Semdeg wonis Sokoladis filebs. qvemoT movcemulia SemTxveviT

SerCeuli Sokoladis 40 filis wonebi gramebSi:

51.5 51.7 51.2 50.4 51.8 49.6 50.6 50.8 50.9 51.7 51.1

51.3 51.0 50.8 51.8 49.3 51.4 51.3 50.7 51.7 50.7 50.8

51.6 52.2 51.2 50.7 50.2 50.8 50.8 51.3 50.4 52.9 51.151.8 49.4 51.5 50.1 51.6 51.5 50.3

9.1. gansazRvreT akmayofilebs Tu ara mwarmoebeli saWiro moTxovnebs,

Tu Sokoladis filis standartuli wona Seadgens 50 grams;

9.2. aageT standartuli wonis 95%-iani ndobis intervali. moicavs

Tu ara es intervali 50 grams?

10. qvemoT moyvanilia studentis mier 18 mgzavrobaSi daxarju-

li dro wuTebSi: 67 73 79 78 50 80 98 88 79 71 51 57

81 65 65 85 65 90. aageT studentis mier mgzavrobaze daxarju-

li drois 95%-iani ndobis intervali.

11. meteorologis azriT qalaqSi qaris saSualo siCqarea 8 km /sT. SemTxveviT SerCeuli 32 dRis monacemebiT qaris saSualo siCqare

aRmoCnda 8.2 km/sT, xolo Sesworebuli standartuli gadaxra ki 0.6

km/sT. 0.05 mniSvnelovnebis doniT gvaqvs Tu ara safuZveli ar da-

veTanxmoT meteorologs? gamoiyeneT P -mniSvnelobis meTodi.

12. dakvirvebebis ra minimaluri raodenobaa saWiro imisaTvis,

rom miRweul iqna 0.05-is toli mniSvnelovnebis done da 0.9-is to-

li simZlavre ( , 49)N normaluri populaciis saSualos Sesaxeb

ZiriTadi 0 : 8H E hipoTezis Semowmebisas 1 : 11H E alternativis

winaaRmdeg.

113

13. gayinuli kerZis mwarmoebeli firmis direqtori acxadebs,rom kerZis saSualo kaloriuloba aris 800, xolo standartuli ga-

daxra ki 25. mkvklevarma Seamowma 12 kerZi da daadgina, rom maTi saS-

ualo kaloriuloba iyo 873. gvaqvs Tu ara sakmarisi safuZveli

0.02 mniSvnelovnebis doniT uarvyoT direqtoris mtkicebuloba?

CavTvaloT, rom kaloriuloba kerZSi ganawilebulia normalurad.

14. dietologis gancxadebiT misi dietiT pacientebi 20 kviris

manZilze saSualod ikleben 24 funts. Sesabamisi standartuli gadax-

raa 5 funti. dietologs surs miiRos ukeTesi Sedegi da amcirebs ma-

rilis moxmarebas. axali meTodis gamoyenebiT 40 SemTxveviT SerCeuli

pacienti 20 kviraSi saSualod iklebs 16.3 funts. 0.05 mniSvnel-ovnebis doniT SeiZleba Tu ara iTqvas, rom dieta Seicvala?

15. pacientebis garkveuli jgufis sisxlSi qolesterinis saSu-

alo done Seadgens aranakleb 240 miligrams, xolo standartuli ga-

daxraa 18 miligrami. axali preparatis miRebis Semdeg SemTxveviT Ser-

Ceul 40 pacientis sisxlSi qolesterinis saSualo done aRmoCnda

229 miligrami. 0.01 mniSvnelovnebis doniT SegviZlia Tu ara

davaskvnaT, rom axali preparati efeqturia?

16. eqimebis azriT morbenali adamiani ufro met Jangbads moixm-

ars vidre saSualod yvela adamiani. SemTxveviT SerCeuli 15 morbena-

lisaTvis Jangbadis moxmarebis saSualo iyo 40.6 ml/kg, xolo Seswo-rebuli standartuli gadaxra ki 6ml/kg. Tu yvela adamianis saSualo

moxmareba Seadgens 36.7ml/kg, gvaqvs Tu ara sakmao safuZveli davuje-

roT eqimebs 0.05 mniSvnelovnebis doniT?

17. ekvatoris samxreTiT zafxulis TveebSi mosuli naleqebis

saSualo Seadgens 11.52 diums (1 diumi = 2.54 sm). 2000 wels mkvlev-

arma SemTxveviT SeraCia ekvatoris samxreTiT mdebare 10 qalaqi da da-

adgina, rom mosuli naleqebis saSualo aris 7.42 diumi. SerCevis Ses-

worebuli standartuli gadaxra Seadgens 1.3 diums. 0.05 mniSvne-

lovnebis doniT SeuZlia Tu ara mkvlevars daaskvnas, rom 2000 wels

mosuli naleqebis saSualo naklebia 11.52 diumze?18. sigaretis kompanias surs Seamowmos hipoTeza, rom mis siga-

retSi nikotinis Semcvlelobis dipersia aris 0.644. nikotinis Semcv-

leloba izomeba miligramebSi da igulisxmeba, rom is normalurad ga-

nawilebulia. 20 sigaretisgan aRebuli SerCevis Sesworebuli standa-

rtuli gadaxraa 1 miligrami. 0.05 mniSvnelovnebis doniT gvaqvs

Tu ara sakmarisi safuZveli uarvyoT kompaniis hipoTeza?

19. dietologis mtkicebiT sxvadasxva saxis erT magidis kovz

sirofSi kaloriebis ricxvis standartuli gadaxra aris 60. 0.1 mniSvnelovnebis doniT SeiZleba Tu ar am mtkicebulebis uaryofa, Tu

SemTxveviT SerCeuli 18 sxvadasxva saxis sirofis erT kovzSi kalor-iebis ricxvia:

114

53 210 100 200 100 220210 100 240 200 100 210

100 210 100 210 100 60

20. kopaniis menejeris mtkicebiT maT mier gamoSvebul iogurt-

Si Saqris Semcvleloba ar aRemateba 25 grams. SemTxveviT SerCeul 20

iogurtSi gazomes Saqris Semcvleloba da aRmoCnda, rom Sesworebuli

SerCeviTi dispersia tolia 36-is. 0.1 mniSvnelovnebis doniT SeiZ-

leba Tu ar am mtkicebulebis uaryofa?

21. samSobiaro saxlis warsuli Canawerebis mixedviT ara dRena-

kluli bavSvebis saSualo wona metia vidre 7 funti da 2 uncia. mim-dinare wels 100 dabadebuli bavSvidan 23-is wona meti iyo vidre 7

funti da 2 uncia. 0.01 mniSvnelovnebis doniT gvaqvs Tu ara sakm-

arisi safuZveli vamtkicoT rom proporcia Seicvala?

22. mkvlevari amtkicebs, rom specialuri dietiT gamokvebili

zrdasruli Rori saSualod iwonis 200 funts. 10 aseTi Rorisagan

Sedgenili SerCevis saSualo wona aRmoCnda 198.2 funti, xolo Seswo-

rebuli standartuli gadaxra ki 3.3 funti. 0.05 mniSvnelovnebis

doniT gvaqvs Tu ara safuZveli davujeroT mkvlevars? aageT 95%-iani

ndobis intervali realuri saSualosaTvis.

23. wina kvlevis Tanaxmad srulwlovani adamianebis sul cota60% sauzmeze miirTmevs kvercxs kviraSi oTxjer mainc. am mosazrebis

Sesamowmeblad dietologma SemTxveviT SearCia 100 srulwlovani adam-

iani da daadgina, rom maT Soris 54% sauzmeze miirTmevs kvercxs kvi-

raSi oTxjer mainc. 0.1 mniSvnelovnebis doniT eTanxmeba Tu ara

dietologis Sedegi wina kvlevis Sedegs?

24. sociologs surs Seamowmos sworia Tu ara, rom garkveuli

profesiis qalebi pirvel Svils aCenen 28.6 wlis asakSi. 0.05 mni-

Svnelovnebis doniT ra daskvna unda gamoitanos sociologma, Tu mas

eqneba SemTxveviT SerCeuli 36 qalis mier pirveli Svilis gaCenis as-

akis Semdegi monacemebi:32 28 26 33 35 34 29 24 22

25 26 28 28 34 33 32 30 29

30 27 33 34 28 25 24 33 25

37 35 33 34 36 38 27 29 26

115

l e q c i a 9.

Tavi 10. hipoTezaTa Semowmeba. oramokrefiani amocanebi.

10.1. Sesavali.

wina leqciaSi Cven vamowmebdiT hipoTezebs erTi populaciis pa-rametrebis konkretuli mniSvnelobis Sesaxeb, romelic SeiZleba iyos

mkvlevaris mier garkveuli mosazrebebiT dasaxelebuli ricxvi, wina

gamokvlevebidan cnobili ricxvi an ufro didi moculobis populacia-

ze dakvirvebebidan miRebuli ricxvi.

axla Cven ganvixilavT e.w. oramokrefiani amocanebis, rac guli-

sxmobs ori populaciis parametrebis Sedarebas, ise, rom arcerTis

konkretuli mniSvneloba cnobili ar aris. sakiTxi ase dgas: mocemu-

lia ori SerCeva raRac populaciebidan, romelTa maxasiaTeblebi (par-

ametrebi) Cven ar viciT da gvainteresebs, aris Tu ara am populacieb-

is saSualoebi (an dispersiebi, an proporciebi) toli. es iqneba CveniZiriTadi (nulovani) hipoTeza, xolo alternativebi aqac SeiZleba iy-

os calmxrivi (marjvena da marcxena) an ormxrivi. Cven xSirad CavTv-

liT, rom Sesadarebeli populaciebi erTmaneTisagan damoukidebelia,

rac imas niSnavs, rom is SemTxveviTi sidideebi, romlebic orive SerC-

evisaTvis ricxviTi monacemebis ukan dgas, arian damoukideblebi.

10.2. dawyvilebuli monacemebi.

magaliTi 10.1. davuSvaT, Cven gvainteresebs moqmedebs Tu ara ga-

rkveuli tipis kontraceptivis miReba qalis sisxlis wnevaze. am mizn-

iT mopovebuli monacemebi warmodgenilia Semdegi cxrilis saxiT:

iwneva

kontraceptivis

gamoyenebamde (xi1)

wneva kontraceptivis

gamoyenebis Semdeg (xi2)

sxvaoba

di = xi2 - xi1

1 115 128 132 112 115 33 107 106 -14 119 128 95 115 122 76 138 145 77 126 132 68 105 109 49 104 102 -210 115 117 2

116

vTqvaT, i-uri qalis sisxlis wneva kontraceptivis miRebamde ga-

nawilebulia normalurad saSualoTi i da dispersiiT 2, xolo kon-

traceptivis miRebis Semdeg agreTve -- normalurad saSualoTi i + da imave 2 dispersiiT. nulovani hipoTeza ase yalibdeba: H0 : = 0,rac imas niSnavs, rom kontraceptivis miRebas wneva ar Seucvlia, xo-

lo alternativaa H1 : 0, rac imas niSnavs, rom kontraceptivis mi-

Reba cvlis wnevis sidides anu moqmedebs sisxlis wnevaze. cxadia, rom

kriteriumis statistika orive SerCevas unda eyrdnobodes. cal-calke

populaciebis parametrebi Cven ar viciT. samagierod, hipoTezis samar-

Tlianobis SemTxvevaSi, CvenTvis cnobilia, rom SemTxveviTi sidideebi

Di Xi2 - Xi1, i =1,2,…,n, ganawilebulia normalurad saSualoTi 0 dadispersiiT, 2

D , romelic marTalia ar viciT, magram romlis Sefaseb-

ac SegviZlia gaerTianebuli SerCevidan. kriteriumis statistikad avi-

RoT

Dn

nn S

DnT

,

(10.1)

SemTxveviTi sidide, sadac

n

iin D

nD

1

1 da Sn,D aRniSnavs sxvaobebis

SerCeviT standartul gadaxras, anu

Sn,D =

2

11

2 )/1(1

1 n

ii

n

ii DnD

n . 10.2)

iseve, rogorc erTamokrefian amocanaSi normaluri populaciisSesaxeb hipoTezis Semowmebisas ucnobi dispersiis dros, Tn SemTxveviT

sidides aqvs t(n-1)-ganawileba da amitom statistikuri kriteriumi aseCamoyalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis Tn SemTxveviT

sididis dakvirvebuli mniSvneloba tn (anu is mniSvneloba, romelic

miiReba Tn SemTxveviTi sididis gamosaxulebaSi Di sididebis Secvlisas

di sididebiT) akmayofilebs pirobas 2/1,12/1,1 nnn ttt , maSin H0

hipoTezis uaryofis safuZveli ara gvaqvs, winaaRmdeg SemTxvevaSi ki

mas uarvyofT.

gavnagrZoT 10.1 magaliTis ganxilva da gamovTvaloT Tn SemTxv-

eviT sididis dakvirvebuli t10 mniSvneloba. cxadia, rom

10d =(13+3-1+9+7+7+6+4+2-2)/10=48/10=4.8;

Dns , = {{[132+32+(-1)2+92+72+72+62+42+22+(-2)2]-

-10(4.8)2}/9}1/2=(20.844)1/2=4.566da amitom t10 = 3.17 4.8 / 4.566 = 3.32. t(9)-ganawilebis kvantilebis

cxrilebidan ki, vpoulobT, rom t9,0.975 = 2.262. radgan t10 = 3.32 mniSvne-

117

loba ar ekuTvnis [-2.262, 2.262] ricxviT intervals, H0 hipoTezas ua-rvyofT. maSasadame, kontraceptivi moqmedebs qalis wnevaze.

sainteresoa rogor intervalSia moTavsebuli sxvaobis WeSmari-

ti saSualo D? ndobis intervalis asagebad viyenebT imave faqts,

romDn

nn S

DnT

,

~ t(n-1) da rogorc me-7 leqciidan viciT, mas eqneba

Semdegi saxe:

1,1 / 2 , 1,1 / 2 ,/ , /n n n D n n n Dd t s n d t s n . (10.3)

amitom Cvens magaliTSi miviRebT:

(4.8-2.2624.566/3.17, 4.8+2.2624.566/3.17) = (1.53, 8.07).maSasadame, saSualo wnevaTa sxvaobis WeSmariti D mniSvneloba

0.95-is toli albaTobiT moTavsebulia 1.53 mm vwy.sv.-dan 8.07 mm vwy.sv.

-mde.

10.3. oramokrefiani t –kriteriumi toli, ucnobi dispersiebis

SemTxvevaSi.

vTqvaT, mocemulia ori damoukidebeli normalurad ganawilebu-

li populacia saSualoebiT 1 da 2 da cnobilia, rom maT aqvT to-

li dispersiebi 21 = 2

2 =2, romelTa saerTo 2 mniSvneloba ucnobia.

Cveni amocanaa SevamowmoT H0 : 1 = 2 hipoTeza H1 : 1 2 alternativis

winaaRmdeg. cxadia, rom21 21 nn XX ~ N(1 - 2, 2(1/n1+1/n2)), sadac n1

da n2 Sesabamisad, pirveli da meore SerCevis moculobebia. Sesabamis-

ad, H0 hipoTezis samarTlianobisas

21

21

/1/121

nn

XX nn

~ N(0,1). 10.4)

magram es sidide kriteriumis statistikad ar gamogvadgeba, ra-

dgan is Seicavs ucnob parametrs. amitom mis magivrad (10.4) gamosa-xulebaSi CavsvaT misi Sefaseba da kriteriumis statistikad aviRoT

miRebuli sidide. ismis kiTxva: romeli SerCevidan SevafasoT ucnobi parametri? am kiTxvaze pasuxis gasacemad, gavixsenoT, rom SerCevis

moculobis zrda iwvevda saSualos standartuli Secdomis Semcireb-

as. amitom iqcevian ase: vinaidan hipoTezis samarTlianobis SemTxvevaSi,

orive populacia erTnairad yofila ganawilebuli, amitom gavaerTian-

oT es SerCevebi (riTac gavzrdiT SerCevis saerTo moculobas) da ise

SevafasoT 2-is mniSvneloba, anu Sefasebad aviRoT Semdegi sidide:

2

)1()1(

21

212

2112

,21

2

nn

SnSnS

nn

nna. (10.5)

118

advili saCvenebelia, rom ase agebuli Sefaseba warmoadgens 2-is gadauadgilebad Sefasebas.

sabolood, kriteriumis statistikas aqvs Semdegi saxe

21,

21,

/1/121

21

21 nnS

XXT

nn

nnnn

, (10.6)

romlisTvisac mtkicdeba, rom

21,

21,

/1/121

21

21 nnS

XXT

nn

nnnn

~ t(n1 + n2 -2). (10.7)

Sesabamisad, statistikuri kriteriumi ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis21 ,nnT SemTxveviT

sididis dakvirvebuli mniSvneloba21 ,nnt akmayofilebs pirobas

2/1,2,2/1,2 212121 nnnnnn ttt , (10.8)

maSin H0 hipoTezis uaryofis safuZveli ara gvaqvs, winaaRmdeg SemTxve-

vaSi ki mas uarvyofT.

kriteriumis gamoyenebis sailustraciod moviyvanoT Semdegi

magaliTi 10.2. davuSvaT, 35-39 ww. asakobrivi jgufis erT daw-esebulebaSi momuSave im 8 qalisaTvis, romlebic xmaroben kontracep-tivs sisxlis wnevaTa saSualo gamovida 132.86 mm vwy.sv., standartu-li gadaxriT 15.34 mm vwy.sv., xolo im 21 qalisaTvis, romlebic ar

xmaroben kontraceptivs, igive maCveneblebma Seadgina Sesabamisad,

127.44 da 18.23 mm vwy.sv.. = 0.05 mniSvnelovnebis doniT ra SeiZleba

iTqvas populaciebis saSualo wnevebs Soris sxvaobaze?

amoxsna. (10.6) formulaze dayrdnobiT gamovTvaloT21 ,nnT SemTx-

veviT sididis dakvirvebuli21 ,nnt mniSvneloba. amisaTvis jer (10.5)-dan

gamovTvaloT 2, 2nna

S -is dakvirvebuli 2, 21 nns mniSvneloba:

2, 21 nns = ((8-1)(15.34)2+(21-1)(18.23)2)/(8+21-2) = 307.18,

saidanac 527.1718.30721 , nns . amitom

21 ,nnt =

21/18/1527.17

44.12786.1320.74.

t(27)-ganawilebis kvantilebis cxrilebidan vpoulobT, rom

t27,0.975 = 2.052. radgan t8,21 = 0.74 mniSvneloba ekuTvnis [-2.052, 2.052]ricxviT intervals, = 0.05 mniSvnelovnebis doniT H0 hipoTezis uar-yofis safuZveli ara gvaqvs. maSasadame, davaskvniT, rom sisxlis saSu-

alo wnevebi mocemuli populaciebisaTvis mniSvnelovnad ar gansxvavde-

ba.

radgan cnobilia kriteriumis statistikis ganawileba, iseve,

rogorc wina nawilSi SegviZlia gamovTvaloT 1- ndobis albaTobis

119

mqone ndobis intervali 1-2 sxvaobisaTvis, romlis qveda da zedasazRvari Sesabamisad iqneba:

1 2 1 2 1 21 1 2,1 / 2 ,1 2

1 1n n n n n nx x t s

n n . (10.9)

10.2 magaliTisaTvis es intervali iqneba:

[132.86-127.44–2.05217.527(1/8+1/21)1/2 ,132.86-127.44–2.05217.527(1/8+1/21)1/2] = [-9.52 , 20.36].

rogorc vxedavT, es intervali sakmarisad farTea da saWiroa

ufro didi moculobis SerCevebi, rom ufro zustad davaxasiaToT 1--2 sxvaobis WeSmariti mniSvneloba.

10.4. hipoTeza ori normaluri populaciis dispersiaTa

tolobis Sesaxeb.

vTqvaT, mocemulia ori SerCeva n1 da n2 moculobebiT, romelTa

Sesabamisi populaciebi normaluradaa ganawilebuli parametrebiT (1,21 ) da (1,

22 ). Sesamowmebelia H0 : 2

1 = 22 hipoTeza H1 : 2

1 22 al-

ternativis winaaRmdeg. rogorc cnobilia, 21 -isa da 2

2 -is gadauadgi-

lebad Sefasebebs warmoadgenen Sesabamisi Sesworebuli SerCeviTi disp-

ersiebi: 21 1nS da 2

2 2nS . magram saSualoebis tolobis Sesaxeb amocanisag-

an gansxvavebiT, 21 1nS da 2

2 2nS -is Sesadareblad ixilaven ara 21 1nS - 2

2 2nS

sxvaobas, aramed maT Sefardebas 21 1nS / 2

2 2nS , romelic gasagebia, rom hip-

oTezis samarTlianobis dros unda Sedardes ara 0-s, aramed 1-s. gar-

da amisa, ucnobia 21 1nS - 2

2 2nS SemTxveviT sididis ganawileba, maSin, roca

21 1nS / 2

2 2nS SemTxveviT sidides hipoTezis samarTlianobis dros aqvs e.w.

fiSeris ganawileba anu F(n1 -1, n2 -1)-ganawileba. statistikuri krite-riumis agebamde davaxasiaToT es ganawileba.

F(1 , 2 )-ganawileba aqvs ori damoukidebeli (normirebuli) 2

SemTxveviTi sididis Sefardebas, romelTa Tavisuflebis xarisxebia Se-

sabamisad, 1 da 2, anu

222

112

, /)(

/)(21

F ~ F(1 , 2 ). (10.10)

2()-ganawilebis msgavsad, es ganawilebac Tavmoyrilia dadebiT

naxevarRerZze. misi pF ,, 21 kvantilebi ganisazRvreba tolobiT:

pFFP p }{ ,,, 2121 . (10.11)

SevniSnoT, rom am ganawilebis (SemTxveviTi sididis) (10.10)

gansazRvridan gamomdinareobs, rom

pF ,, 21 = 1/ pF 1,, 12 . (10.12)

120

am Tvisebis gamo, cxrilebSi F-ganawilebis mxolod zeda kriti-kuli mniSvnelobebia mocemuli.

davubrundeT axla statistikur kriteriums, romelsac mokled

F-kriteriums uwodeben. is ase yalibdeba:

Tu mocemuli mniSvnelovnebis donisaTvis21 ,nnF = 2

1 1nS / 22 2nS

statistikis dakvirvebuli mniSvneloba21 ,nnf = 2

1 1ns / 22 2ns akmayofilebs

pirobas, rom

2/1,1,1,2/,1,1 212121 nnnnnn FfF , (10.13)

maSin mniSvnelovnebis doniT H0 : 21 = 2

2 hipoTezis uaryofis safu-

Zveli ara gvaqvs, winaaRmdeg SemTxvevaSi ki mas uarvyofT.

kriteriumis gamoyenebis sailustraciod ganvixiloT Semdegi

magaliTi 10.2. davuSvaT, 10.1 magaliTSi CvenTvis ucnobia dispe-

rsiaTa tolobis piroba. rogorc gvaxsovs, 21 qalis (romlebic ar

iyeneben kontraceptivs) sisxlis wnevis standartuli gadaxra toli

iyo 18.23 mm vwy. sv., xolo 8 qalis (romlebic iyeneben kontracept-ivs) sisxlis wnevis standartuli gadaxra toli iyo 15.34 mm vwy.sv.-is. = 0.05-saTvis SevamowmoT hipoTeza imis Sesaxeb, rom am populac-

iebis dispersiebi tolia.

amoxsna. gamovTvaloT 8,21F statistikis dakvirvebuli mniSvnel-

oba: 8,21f =18.232/15.342=1.41. F-ganawilebis cxrilebidan vpoulobT,

rom F20,7,0.975F24,7,0.975=4.42>1.41. amitom mocemuli mniSvnelovnebis do-

niT dispersiaTa tolobis Sesaxeb hipoTezis uaryofis safuZveli ara

gvaqvs.

10.5. oramokrefiani t –kriteriumi aratoli dispersiebis

SemTxvevaSi.

vTqvaT, mocemulia ori damoukidebeli normalurad ganawilebu-

li populacia saSualoebiT 1 da 2 da cnobilia, rom maT aqvT ara-

toli dispersiebi 21 2

2 . Cveni amocanaa SevamowmoT H0 : 1 = 2 hipo-

Teza H1 : 1 2 alternativis winaaRmdeg. iseve, rogorc 9.3 punqtSi,

ganvixiloT21 21 nn XX sxvaoba, romlisaTvisac cxadia, rom

21 21 nn XX ~ N(1 - 2; (21 /n1+

22 /n2)),

sadac n1 da n2 Sesabamisad, pirveli da meore SerCevis moculobebia.

maSasadame, Tu cnobilia 21 da 2

2 , maSin H0 hipoTezis samarTlianobi-

sas

2221

21

21

//

21

nn

XX nn

~ N(0,1) (10.14)

121

da es SemTxveviTi sidide gamodgeba kriteriumis statistikad.magram es sidide kriteriumis statistikad ar gamogvadgeba, ro-

ca is Seicavs ucnob 21 da 2

2 parametrebs. amitom (iseve rogorc 9.3

punqtSi) ucnobi 21 -isa da 2

2 -is (10.14) gamosaxulebaSi vsvamT maT

Sefasebebs (Sesabamis SerCeviT standartuli gadaxrebs) da kriterium-

is statistikad viRebT Semdeg SemTxveviT sidides:

2221

21

21,

//21

21

21

nSnS

XXT

nn

nn

nn

. (10.15)

rogorc viciT, kriteriumis statistika ara mxolod gamoTvla-

di sidide unda iyos, aramed kriteriumis asagebad, arsebiTia misi zu-

sti an asimptoturi ganawilebis codnac. rogoraa ganawilebuli

21 ,nnT SemTxveviTi sidide? maTematikur statistikaSi es problema cno-

bilia berens-fiSeris problemis saxeliT. misi gadaWris erT-erTi, Se-

darebiT martivi gza cnobilia satertvaitis (Satterthwaite) meTodissaxeliT, romelic Semdegnairad yalibdeba:

Tu mocemuli mniSvnelovnebis donisaTvis21 ,nnT statistikis

dakvirvebuli mniSvneloba21 ,nnt akmayofilebs Semdeg utolobas

-t[c] , 1-/2 21 ,nnt t[c] , 1-/2, (10.16)

sadac c gamoiTvleba Semdgi wesiT

)1/()/()1/()/(

)//(

22

2221

21

21

22

221

21

21

21

nnsnns

nsnsc

nn

nn, (10.17)

([c] aRniSnavs c ricxvis mTel nawils anu c-s uaxloes mTel ricxvs

marcxnidan), maSin mniSvnelovnebis doniT H0 : 1 = 2 hipoTezis uar-

yofis safuZveli ara gvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT.

kriteriumis gamoyenebis sailustraciod ganvixiloT SemdegimagaliTi 10.3. vTqvaT, Seiswavleba qolesterinis Semcveloba im

adamianebis memkvidreebSi, romlebic gardaicvalnen gulis daavadebiT.

davuSvaT, 100 aseT bavSvze dakvirvebam mogvca qolesterinis saSualo

done 207.3 mg/dl, xolo standartuli gadaxra ki 35.6 mg/dl. garda

amisa, daakvirdnen qolesterinis doneebs im bavSvebSi, romelTa mSobl-

ebic cocxlebia da ara aqvT gulis daavadeba. SeirCa aseTi 74 bavSvi,romlis mixedviTac qolesterinis saSualo done gamovida 193.4 mg/dl,

xolo standartuli gadaxra ki 17.3 mg/dl. Cveni amocanaa = 0.05 mni-Svnelovnebis donisaTvis SevamowmoT hipoTeza populaciebis saSualoe-

bis tolobis Sesaxeb.

amoxsna. SegviZlia davuSvaT, rom populaciebi ganawilebulia

normalurad, Tumca maTi parametrebis Sesaxeb araferia cnobili. amit-

om es amocana gaiyofa or etapad: I. jer SevamowmoT dispersiebis to-

lobis hipoTeza 10.4 punqtis Sesabamisad. II. Tu aRnSinul hipoTezas

122

ver uarvyofT, Semdeg gamoviyenoT 10.3 punqtSi miRebuli kriteriumisaSualoTa tolobis Sesaxeb hipoTezis Sesamowmeblad, xolo Tu ar

dadasturda dispersiaTa tolobis hipoTeza, saSualoTa tolobis hip-

oTeza SevamowmoT am punqtSi ganxiluli satertvaitis meTodiT.

ganvixiloT SerCeviT dispersiaTa Sefardeba da SevadaroT is

F99,73,0.975 kvantilis ricxviT mniSvnelobas. 74,100F statistikis dakvirv-

ebuli mniSvneloba tolia 74,100f = 35.62/17.32 = 4.23. radgan cxrilebSi

ar aris F99,73,0.975 kvantilis ricxviTi mniSvneloba, gamovTvaloT p-mni-Svneloba, romelic alternativis ormxrivobis gamo tolia p = 2 P{ 74,100F 4.23} 0.0002, rac imis maCvenebelia, rom Sedegi statistiku-

rad mniSvnelovania da maSasadame, hipoTeza dispersiaTa tolobis Sesa-

xeb unda uarvyoT.

gadavideT meore etapze. radgan davadgineT, rom populaciaTa

dispersiebi gansxvavebulia, viyenebT satertvaitis meTods saSualoTa

tolobis Sesaxeb hipoTezis Sesamowmeblad. gamovTvaloT (10.15) tol-

obiT gansazRvruli statistikis dakvirvebuli mniSvneloba:

4.3089.4

9.13

74/3.17100/6.35

4.1933.2072274,100

t .

gamovTvaloT axla (10.17) Tanafardobis mixedviT Tavisuflebis

xarisxis miaxloebiTi mniSvneloba c:

4.1518465.1

718.16

73/)74/3.17(99/)100/6.35(

)74/3.17100/6.35( 2

222

222

c

da maSasadame, [c] = 151. radgan t100,74 = 3.4 > t120, 0.975 = 1.98 > t151, 0.975 ami-

tom = 0.05 mniSvnelovnebis doniT populaciaTa saSualoebis tolob-is hipoTezas uarvyofT.

SevniSnoT, rom 1- ndobis albaTobis mqone ndobis intervals

aqvs Semdegi saxe:

1 2 1 2

1 2 1 2

2 2 2 21 2 1 2

1 1 [ ],1 / 2 1 1 [ ],1 / 21 2 1 2

,n n n nn n c n n c

s s s sx x t x x t

n n n n

. (10.18)

10.6. SerCevaTa moculobebis gansazRvra. ori populaciis

saSualoebis Sedarebis kriteriumis simZlavre.

SerCevaTa moculobebis gansazRvra, rogorc erTamokrefian amo-

canebSic vnaxeT, saWiroa eqsperimentis winaswar dagegmvisaTvis. am na-

wilSi Cven moviyvanT oramkrefiani amocanebisaTvis SerCevaTa im mocu-

lobebis formulebs, romlebic mocemuli mniSvnelovnebis donisaTv-is populaciaTa saSualoebis Sedarebis amocanaSi saWiroa dasaxelebu-

li 1- simZlavris misaRwevad ormxrivi alternativebis dros.

123

davuSvaT, mocemulia ori normaluri populacia saSualoebiT

1 da 2 da cnobili dispersiebiT 21 da 2

2 . Sesamowmebelia H0 : 1=

= 2 hipoTeza H1 : 1 2 alternativis winaaRmdeg. Cveni amocanaa, ganv-

sazRvroT populaciaTa is n1 da n2 = kn1 moculobebi, romelTaTvisac

mniSvnelovnebis donisaTvis miiRweva mocemuli 1- simZlavre. form-ulebs aqvT Semdegi saxe:

212

212/1

22

211 )/()()/( zzkn , (10.19)

212

212/1

22

212 )/()()( zzkn . (10.20)

SevniSnoT, rom k=1 SemTxvevaSi n1 = n2 da (10.19) da (10.20) Se-

fasebebic iZleva erTsa da imave Sedegs. sailustraciod ganvixiloT

10.2 magaliTis Semdegi modifikacia.

magaliTi 10.4. davuSvaT, 35-39 ww. asakobrivi jgufis im qalebs

Soris, romlebic xmaroben kontraceptivs, sisxlis wnevis saSualoa132.86 mm vwy.sv. da standartuli gadaxra ki 15.34 mm vwy.sv., xolo

qalebisaTvis, romlebic ar xmaroben kontraceptivs, igive maCveneblebi

Sesabamisad Seadgens 127.44-sa da 18.23 mm vwy.sv.-s. = 0.05 mniSvnelo-

vnebis donisa da k=2-saTvis ganvsazRvroT SerCevaTa is moculobebi,

romlebic saSualoTa garCevis amocanaSi mogvcemen 80%-ian simZlavr-

es?

amoxsna. (10.19) formulidan miviRebT, rom

1.107)44.12786.132(

)84.096.1()2/23.1834.15(2

222

1

n ,

saidanac davaskvniT, rom n1 = 108 da maSasadame, n2 = 2 n1 = 2 108 = 216.kriteriumis simZlavris gamosaTvlelad H0 : 1 = 2 hipoTezis

winaaRmdeg ganvixiloT specifikuri alternativa, kerZod, Semdegi al-

ternativa H1 : 2 = 1 + . maSin ormxrivi alternativisaTvis simZlav-re Semdegnairad iTvleba:

k

nz

/1

22

21

12/1

, sadac k = n2 / n1. (10.21)

magaliTi 10.5. SeirCa 35-39 ww. asakobrivi jgufis 100-100 qa-li, romlebic xmaroben da ar xmaroben kontraceptivs da sisxliswnevaTa sxvaobis saSualo gamovida 5 mm vwy.sv.. vTqvaT, standartuli

gadaxrebi am populaciebSi Sesabamisad 15.34 mm vwy.sv.-sa da 18.23 mmvwy.sv.-is tolia. = 0.05 mniSvnelovnebis donisaTvis ganvsazRvroT

kriteriumis simZlavre.

amoxsna. radgan n1 = n2 = 100, k = n2 / n1=1, = 5, 1 = 15.34, 2 =18.23 da = 0.05 (10.21) tolobidan miviRebT, rom

1- = (-1.96+2.099) = (0.139) = 0.555.

124

amocanebi1. 12-14 wlis dabal socialur-ekonomikur pirobebSi mcxovrebi

25 gogonas kalciumis miRebis saSualo da standartuli gadaxra Ses-

abamisad tolia 6.56-isa da 0.64 mg-is. ukeTes pirobebSi mcxovrebi 40imave asakis gogonasaTvis ki msgavsi maCveneblebi Sesabamisad Seadgens

6.8-sa da 0.76 mg-s. davuSvaT, = 0.05.1.1. SeamowmeT, aris Tu ara mniSvnelovani gansxvaveba ori populaciis

dispersiebs Soris;

1.2. rogoria saSualoebis Sedarebis Sesaferisi procedura?

1.3. CaatareT es procedura ornairad: kritikul mniSvnelobasa da p-mniSvnelobaze dayrdnobiT;

1.4. gamoTvaleT 95%-iani ndobis intervali populaciebis saSualoTa

sxvaobisaTvis;

1.5. davuSvaT, Semdgomi SeswavlisaTvis saWiroa dadgindes ori jgufisgogonaTa Tanabari raodenoba, imisaTvis, rom mivaRwioT 80%-ian simZ-

lavres = 0.05 mniSvnelovnebis donisaTvis. ramdeni monacemia amisaTv-

is saWiro?

1.6. rogor Seicvleba pasuxi 1.5 amocanaSi, Tu aviRebT = 0.01-s?2. safiqrebelia, rom bavSvis filtvebis janmrTelobas gansazRv-

ravs ojaxSi ewevian Tu ara sigarets. amis Sesaswavlad SeirCa 5-9

wlis asakis bavSvebis ori jgufi: 23 im ojaxebidan, sadac ar eweviansigarets da 20 im ojaxebidan, sadac sigarets ewevian. aRmoCnda, rompirvel jgufSi FEV (forced expiratory volume) sididis saSualom da

standartulma gadaxram Sesabamisad Seadgina 2.1 l da 0.7 l, xolo

meore jgufisaTvis igive maCveneblebma Sesabamisad Seadgines 2.3 l da

0.4 l. davuSvaT, = 0.05.2.1. rogor CamoayalibebdiT nulovan da alternatiul hipoTezebs?

2.2. rogoria 2.1-Si Camoyalibebuli hipoTezebis Semowmebis saWiro

procedura?

2.3. CaatareT es procedura kritikul mniSvnelobaze dayrdnobiT;

2.4. gamoTvaleT 95%-iani ndobis intervali populaciebis saSualoTa

sxvaobisaTvis;

2.5. davuSvaT, Semdgomi SeswavlisaTvis saWiroa dadgindes ori jguf-

is bavSvebis Tanabari raodenoba, imisaTvis, rom mivaRwioT 95%-ian si-

mZlavres = 0.05 mniSvnelovnebis donisaTvis. ramdeni monacemia amisa-Tvis saWiro?

3. axseniT, ratom ar SeiZleba ubralod SexedoT eqsperimentul

pirobebSi myofi ori jgufis saSualoTa mniSvnelobebs Soris gansxva-

vebas da gadawyvitoT, aris Tu ara es gansxvaveba ganpirobebuli faqt-

oris moqmedebiT.

125

4. aqvs Tu ara diazepams (sedatiuri saSualeba) gverdiTi efeq-ti? am hipoTezis Sesamowmeblad, 12 pacients aZlevdnen placebos, me-

ore jgufis 12 pacients ki 15mg diazepams. Oorive jgufi afasebda xa-

nmokle Sualedis xangrZlivobas. Pplacebo jgufma es dro Seafasa

rogorc 19.8wm. (standartuli gadaxriT 7.6wm), maSin roca eqsperimen-

tuli jgufis mier Sefasebuli dro iyo 13.6wm. (standartuli gadax-

riT 4.8wm.). aris Tu ara saSualoTa Soris mniSnelovani gansxvaveba?

5. ganapirobebs Tu ara kompiuterulad modelirebuli eqsperim-

entebi ufro did miRwevebs biologiaSi, vidre laboratoriuli eqspe-

rimentebi? 28 studenti-biologi gadaanawiles or jgufSi. pirvel

jgufSi iyo Cveulebrivi swavleba, meoreSi ki kompiuterulad model-irebuli. kursis bolos biologiis codna testiT gazomes:

meTodi Sefaseba

Cveule-

brivi

74 75 81 90 67 72 84 87 92 70 78 82 75 85

kompiu-

teruli

84 75 94 88 69 90 86 97 81 91 83 92 82 83

mniSnelovnad gansxvavdeba Tu ara miRwevis testis mocemuli qulebierTmaneTisagan 0.05 mniSvnelovnebis doniT? gamoiyeneT ormxrivi kri-

teriumi.

6. cxrilSi mocemulia 20-24 wlis qalebis wonebi (funtebSi)

mowevisaTvis Tavis danebebis programis dasawyisSi da mowevisaTvis Ta-

vis danebebidan 1 wlis Semdeg:

per-

iodi

wona x ns

dasaw-

yisSi

122.4 117.1 127.8 123.2 112.5 118.5 121.4 131.7 121.85 6.03

Semdeg 127.9 119.4 127.1 126.5 114.6 125.4 119.8 134.2 124.4 6.13

statistikurad mniSnelovnad gansxvavdeba Tu ara am or periodSi gaz-

omili wonebi erTmaneTisagan?

7. aqvs Tu ara xmaurs raime gavlena wnevaze? 10 studenti isv-

enebda komfortul savarZlebSi 30 wuTis ganmavlobaSi. amis Semdeg

maT uzomavdnen sisxlis sistolur wnevas. meore jgufis 10 studenti

savarZlebSi 30 wuTi usmenda didi qalaqis transportis Canawers pik-

is saaTebSi da maT mere uzomavdnen sisxlis sistolur wnevas.

jgufi sisxlis sistoluri wneva

uxmauro 106 117 124 129 115 131 121 115 128 136

xmauriani 141 136 124 139 121 119 147 128 115 134

a) TiToeuli jgufisaTvis ipoveT saSualo da standartuli gadaxra;

b) ipoveT 95%-ani ndobis intervali ori populaciis saSualoTa So-

126

ris sxvaobisaTvis; g) upasuxeT kiTxvas: moqmedebs Tu ara xmauri sis-xlis sistolur wnevaze.

8. Sedarebuli iyo wesrigis damrRvevad cnobili studentebis

akademiuri moswrebis qulebi im studentebis akademiur moswrebas, ro-

mlebic ar iyvnen SeniSnuli wesrigis darRvevaSi. akademiuri moswrebis

qulis saSualo da standartuli gadaxra wesrigis damrRvevi 1794

studentisaTvis Sesabamisad iyo 1.68 da 0.34. 3606 studentisaTvis ki,

romlebic ar arRvevdnen wesrigs, moswrebis saSualo da standartuli

gadaxra iyo Sesabamisad 2.09 da 0.83. mniSvnelovnad gansxvavdeba Tu

ara moswrebis qulebi? gamoiyeneT ormxrivi kriteriumi da mniSnelov-

nobis 0.05 done.9. mravali studenti gamocdis dros ganicdis SfoTvas, romel-

ic Zalian uSlis xels maT akademiur moswrebas. wamoayenes hipoTeza,

rom damamSvidebeli musikis mosmena gamocdisas Seamcirebs am SfoTvas.

Terapevtma airCia 34 studenti, vinc xasiaTdeboda Zlieri SfoTviT

gamocdis dros. studentebi SemTxveviTad iyvnen gadanawilebuli or

jgufSi. pirveli jgufis studentebi atarebdnen erT saaTs wynar das-

asvenebel oTaxSi, sadac isini kiTxulobdnen Jurnalebs. meore, eqspe-

rimentuli jgufis pirebi ki erT saaTis ganmavlobaSi gamocdis win

usmendnen damamSvidebel musikas. gamocdis qulis saSualo pirveli

jgufisaTvis iyo 73.2, standartuli gadaxra ki 8.4. meore jgufis sa-Sualo Seadgenda 75.6-s, standartuli gadaxra ki 8.7. gansxvavdebian

Tu ara gamocdis qulebi am jgufebSi erTmaneTisagan? gamoiyeneT orm-

xrivi kriteriumi da mniSvnelovnobis 0.05 done.

10. gazomes 85 aRgznebuli pacientis Ramis Zilis xangrZlivoba.

saSualo gamovida 7.16 saaTi, standartuli gadaxra ki 2.15 saaTi. 113

mSvidi pacientis Ramis Zilis saSualo xangrZlivobaa 7.82, standart-

uli gadaxra ki 2.19. gansxvadeba Tu ara erTmaneTisagan am jgufebis

saSualoebi. gamoiyeneT 0.03 done.

11. moqmedebs, Tu ara dedebis SfoTva bavSvebis qcevaze? 48 sami

wlis bavSvis saSualo qcevis qula, romelTa dedebis SfoTvis doneSefasebuli iyo rogoc maRali gamovida 7.92, standartuli gadaxra

ki 3.45. mSvidi dedebis 48 bavSvis saSualo qulaa 5.80, standartu-

li gadaxra ki 2.87.

11.1 SeamowmeT es hipoTeza 0.05 mniSvnelovnebis doniT;

11.2 aageT 90%-iani ndobis intervali saSualoTa sxvaobisaTvis.

12. akvirdebodnen 10 epilefsiiT daavadebul pacients, romelT-

ac utardebodaT standartuli mkurnaloba da 13 pacients, romelTac

utardebodaT mkurnaloba axali medikamenturi preparatebiT da orive

SemTxvevaSi zomavdnen Setevis xangrZlivobas:

standartulimkurnaloba

axalimkurnaloba

127

Setevebis saSualo (wm) 4.1 2.6

standart. Ggadaxra 1.7 0.6

SerCevis moculoba 10 13

12.1 amoklebs Tu ara axali mkurnaloba Setevis xangrZlivobas?

13. epilefsiiT daavadebul 13 Tkufis cali dayofili iyo or

jgufad. Ppirvel jgufs utardebodaT standartuli mkurnaloba, meo-

re jgufis wevrebi iRebdnen axal wamals. qvemoT moyvanilia analizis

Sedegebi.Zveli mkurnaloba 3.9 4.1 3.8 4.4 4,8 4.0 3.7 4.2 4.7 3.5

3.2 4.1

axali mkurnaloba 2.4 2.6 2.4 2.8 2.9 2.5 2.1 2.7 2.9 1.9

1.1 2.5

13.1 aris Tu ara gansxvaveba analizis Sedegebis or simravles Soris?

13.2 SeamowmeT hipoTeza 0.05 mniSvnelovnobis doneze.

14. filtvis kiboTi daavadebul 23 pacients utardeboda stan-

dartuli mkurnaloba, 19-s ki eqsperimentuli. cxrilSi moyvanilia am

pacientebis SerCeviTi maxasiaTeblebi:

eqsperimentalurimkurnaloba

standartulimkurnaloba

saSualo 48.11 37.35

standartuli gadaxra 14.36 19.09

SerCevis moculoba 19 23

aris Tu ara gansxvaveba sicocxlis xangrZlivobaSi? aiReT 0.05 mniSn-

elovnobis done.

15. 100 bayayi iyo gadayvanili erTi sacxovrebeli adgilidan me-

ore adgilzeA(am jiSis bayayebi 30 wlamde cocxloben da sicocxlisbolomde izrdebian). erTi wlis Semdeg gazomili iyo orive adgilas

daWerili bayayebi:

moculoba saSualo wona stand. gadaxra

Zveli adgili 44 1.901 1.468

Aaxali adgili 7 3.95 2.161

SeiZleba Tu ara rom davaskvnaT, rom bayayebisaTvis axali adgili uke-

Tesia?16. cxrilSi moyvanilia erTi da igive wels dabadebuli deda-

li da mamali meliebis wona.

Ddedali meliebis wona mamali meliebis wona

2200 2250 1900 1800 2000 1925 1700 1850 1750

1925 1650 2000 2200 1750 2650 2550 1850 1800

1950 1800 1000 17500 2300 2350 2275 2300 2650

aris Tu ara gansxvaveba meliebis wonaSi ( 0.05 )?

128

16. qveviT mocemulia erTi da igive asakis, sqesis da danaSaul-ebrivi istoriis mqone damnaSaveebis monacemebi. kerZod ki mocemulia

drois xangrZlivoba damnaSaveebis cixidan ganTavisuflebasa da maT mi-

er danaSaulis xelaxla Cadenas Soris. danaSaulis xelaxla Cadenamde

dro aRwerilia TveebiT. Mmocemulia damnaSaveebis ori jgufi, romle-

bic gansxvavdebian erTmaneTisagan patimrobis misjili tipis mixedviT.

cixe savaldebulo

muSaobasTan erTad 14 3 16 10 11 12

mxolod cixe 18 15 21 14 16

16.1. ipoveT TiToeuli jgufis saSualo;

16.2. saSualoTa Soris arsebuli gansxvaveba statistikurad mniSvnel-ovania, Tu aramniSvnelovania? 0.05 .

17. Semowmebul iqna gafantuli skleroziT daavadebul adamian-

Ta ori 120 da 34 kaciani jgufis unarebi. pirvel jgufSi Sediodnen

is adamianebi visac uvlida sakuTari meuRle, xolo meoreSi ki is ad-

amianebi visac uvlida ucxo adamiani. miRebuli Sedegebia: 120 2x ,'1 0.6s ; 34 1.7y , '

2 0.7s . 0.1 mniSvnelovnebis doniT aris Tu ara

gansxvaveba am ori jgufis saSualoebs Soris?18. mkvlevars surs gaarkvios aris Tu ara mweveli adamianis

pulsi ufro maRali vidre aramwevelis. SemTxveviT SerCeuli 100 mwe-

velisa da 100 aramwevelis gamokvlevis Sedegebia Sesabamisad:

100 90x , '1 5s ; 100 88y , '

2 6s . SeuZlia Tu ara mkvlevars 0.05 mniSvnelovnebis doniT daaskvnas, rom mweveli adamianis pulsi ufro

maRalia vidre aramwevelis?

19. ori sxvadasxva dasaxelebis sigaretis SemTxveviT SerCeul30 – 30 RerSi nikotinis Semcvlelobis donis mixedviT gamoTvlili

Sesabamisi SerCeviTi maxasiaTeblebia: 30 28.6x , '1 5.1s ; 30 32.9y ,

'2 4.4s . aageT 99%-iani ndobis intervali saSualoTa sxvaobisaTvis.

20. jandacvis saministros monacemebiT dazRveuli (Sesabamisad,

dauzRvevuli) mSobiare qalebi samSobiaroSi rCebian saSualod 2.3

dRis (Sesabamisad, 1.9 dRis) ganmavlobaSi. 16 - 16 mSobiare qalisagan

Sedgenili ori jgufis samSobiaroSi gatarebuli dReebis Sesworebu-

li standartuli gadaxra qalebis orive kategoriisaTvis Seadgens 0.6

dRes. 0.01 mniSvnelovnebis doniT SeamowmeT hipoTeza saSualoTa

tolobis Sesaxeb. aageT 99%-iani ndobis intervali saSualoTa sxvao-

bisaTvis. isargebleT P -mniSvnelobis meTodiT.

21. qvemoT moyvanilia dro, romelic sWirdeba 6 – 6 TeTr dayavisfer Tagvs, raTa iswavlos martiv labirinTSi garbena. 0.05 mniSvnelovnebis doniT gaarkvieT iwvevs Tu ara Tagvis feri labiri-

129

nTSi garbenis Sesaswavlad saWiro droSi gansxvavebas? aageT 95%-ianindobis intervali saSualoTa sxvaobisaTvis.

TeTri Tagvi 18 24 20 13 15 12

yavisferi Tagvi 25 16 19 14 16 10

22. mkvlevars ainteresebs aris Tu ara gansxvaveba mweveli da

aramweveli adamianebis pulsis ricxvTa dispersiebs Soris. SemTxveviT

SerCeuli 26 mwevelisagan da 18 aramwevelisagan Semdgari ori SerCe-

vis Sesabamisi Sesworebuli SerCeviTi dispersiebia: '21 36s , '2

2 10s .

0.05 mniSvnelovnebis doniT gvaqvs Tu ara sakmarisi safuZveli da-

vaskvnaT, rom dispersiebi gansxvavebulia? CaTvaleT, rom sidide ganaw-

ilebulia normalurad.

23. ginekologis mtkicebiT axalSobili vaJebis simaRleTa cva-

lebadoba gansxvavdeba axalSobili gogonebis simaRleTa cvalebadobis-

agan. SemTxveviT SerCeuli 15 axalSobili vaJis simaRleTa Sesworebu-

li standartuli gadaxra aRmoCnda 1.3 diumi (1 diumi = 2.54 sm), xo-

lo SemTxveviT SerCeuli 15 axalSobili gogonasaTvis ki 0.9 diumi.0.1 mniSvnelovnebis doniT SeiZleba Tu ara davadasturoT gineko-

logis mtkicebuleba?

24. mkvlevaris mtkicebiT arteriuli wnevis cvalebadoba Warbw-

onian adamianebSi ufro didia vidre normalurwonian adamianebSi. SemT-

xveviT SerCeul 28 Warbwonian adanmianSi arteriuli wnevis Sesworeb-

uli standartuli gadaxra aRmoCnda 6.2 mm. vercx. sv., xolo 25 nor-

malurwonian adamianSi ki 2.7 mm. vercx. sv.. 0.01 mniSvnelovnebis

doniT SeamowmeT mkvlevaris mtkicebuleba.

25. mkvlevars surs Seafasos ori popularuli A da B diet-

is Sedegad pacientTa mier dakarguli wonis dispersia. A dietis mim-devari SemTxveviT SerCeuli 10 adamianis mier erT TveSi dakarguli

wonebis Sesworebuli standartuli gadaxra aRmoCnda 6.3 funti, xo-

lo B dietis mimdevari SemTxveviT SerCeuli 12 adamianis ki 4.8 fu-

nti. 0.05 mniSvnelovnebis doniT, SegviZlia Tu ara davaskvnaT,

rom A dietis SemTxvevaSi dakarguli wonis dispersia ufro didia

vidre B dietis SemTxvevaSi?

l e q c i a 10.

Tavi 11. mravalamokrefiani amocanebi. dispersiuli analizi.

11.1. Sesavali.

wina leqciaSi laparaki iyo ori normalurad ganawilebuli po-

pulaciis saSualoebis Sedarebis amocanebze. xSirad saWiroa ufro me-

ti populaciis saSualoebis Sedareba.

130

magaliTi 11.1. janmrTelobis dacvis saministros ainteresebdaaxdens, Tu ara, gavlenas da ramdenad, sigaretis moweva filtvebis

funqciurobaze. amisaTvis filtvebis funqciuroba, anu FEF (forced mid-expiratory flow) sididis mniSvnelobebi gazomil iqna adamianebis eqvs

jgufSi: 1) aramwevelebi (pirobiTad, NS); 2) pasiuri mwevelebi (PS);3) mwevelebi, romlebic filtvebSi ar uSveben kvamls (NI); 4) sustimwevelebi (dReSi 1-dan 10 Reramde 20 an meti wlis ganmavlobaSi,

LS); 5) zomierad mwevelebi (dReSi 11-dan 39 Reramde 20 an meti

wlis ganmavlobaSi, MS) da 6) magari mwevelebi (40 da meti Reri

dReSi, 20 an meti wlis ganmavlobaSi, HS). saiteresoa FEF-is saSua-loebis Sedareba am eqvs jgufSi. Sedegebi mocemulia cxrilSi:

jgufis nomeri

da saxeli

FEF-is saSualo da

standartuli gadaxra (l/w)

SerCevis

moculobebi

1 NS 3.78 0.79 2002 PS 3.30 0.77 2003 NI 3.32 0.86 504 LS 3.23 0.78 2005 MS 2.73 0.81 2006 HS 2.59 0.82 200

t-kriteriumis meTodologia, sakmarisad kargad zogaddeba aseTiSemTxvevebisaTvis da am meTodologias saxelad hqvia erTfaqtorianidispersiuli analizi (SemoklebiT, dispersiuli analizisaTvis xmaro-

ben ANOVA-s, Sesabamisi inglisuri terminis Semoklebidan – Analysisof variance). am tipis amocanebSi mocemulia ni, i =1,2,…,k, moculobis kSerCeva (jgufi); aRvniSnoT yij-Ti i-uri jgufis j-uri monacemi, roml-

is ukan mdgomi Yij SemTxveviTi sididisaTvis vuSvebT, rom samarTliania

Semdegi modeli:

Yij = + i + Xij, (11.1)

sadac mudmivi ricxvia, i aris i-uri jgufis maxasiaTebeli faqto-

ri (efeqti), romelic SeiZleba iyos deterministuli (araSemTxveviTi)

an SemTxveviTi da Xij aris SemTxveviTi sidide, romelsac Secdomas

uwodeben da romelic ganawilebulia normalurad saSualoTi 0 da

dispersiiT 2. iTvleba agreTve, rom 01

k

ii da maSasadame, Yij SemTx-

veviTi sidide ganawilebulia normalurad saSualoTi + i da dispe-

rsiiT 2. aseT pirobebSi amboben, rom (11.1)-iT gansazRvrulia erTfaq-toriani ANOVA modeli. SevniSnoT, rom aseT amocanebSi adgili aqvs

ori tipis variaciulobas: variaciulobas jgufebs Soris da variaciu-

lobas jgufebis SigniT. (11.1)-iT gansazRrul modelSi gaiazreba,rogorc yvela jgufis gaerTianebiT miRebuli SerCevis Sesabamisi pop-

ulaciis saSualo. i sididis azri is aris, rom is warmoadgens gans-

131

xvavebas i-uri jgufis saSualosa da saerTo saSualos Soris, dabol-

os, Xij warmoadgens i-uri jgufis + i saSualodan SemTxveviT gadax-

ras (Secdomas).

intuiciurad, 11.1 magaliTSi FEF-is sidide SeiZleba gaviazroT

rogorc FEF-is saerTo saSualo sidides damatebuli mwevelTa TiTo-

euli jgufis efeqti da damatebuli SemTxveviTi variaciuloba TiToe-

uli jgufis SigniT.

11.2. hipoTezaTa Semowmeba erTfaqtorian ANOVA modelSi.

deterministuli efeqtebis SemTxveva.

nulovani hipoTeza am amocanaSi yalibdeba ase: yvela jgufs

aqvs erTnairi saSualo, anu radgan Cven davuSviT, rom 01

k

ii , nul-

ovani hipoTeza iqneba: H0 : 1 = 2 =…= k = 0, xolo alternativa as-

eTia: H1 : arsebobs i, i = 1, 2,…, k, rom i 0. aRvniSnoT

in

jij

ii y

ny

1

1

simboloTi i-uri jgufis SerCeviTi saSualo da

k

iii yn

ny

1

1 simbo-

loTi yvela monacemis saerTo saSualo, sadac n = n1 + n2 +…+ nk, cxa-

dia, aris yvela monacemis gaerTianebiT miRebuli SerCevis moculoba.

individualuri monacemis gadaxra saerTo saSualosagan, SeiZleba ganv-

ixiloT, rogorc ori gadaxris jami, romlidanac pirveli gviCvenebs

individualur gadaxras jgufuri saSualosagan, xolo meore ki jguf-is saSualos gadaxras saerTo saSualosagan:

yij - y = ( yij - iy ) + ( iy - y ). (11.2)

amitom cxadia, rom Tu (11.2)-is orive mxares aviyvanT kvadrat-Si, avjamavT (orive mxares) yvela i da j-Ti da gaviTvaliswinebT, rom

marjvena mxaris Sereuli wevrebis jami 0-is tolia, miviRebT:

k

i

n

jij

i

yy1 1

2)( =

k

i

n

jiij

i

yy1 1

2)( +

k

i

n

ji

i

yy1 1

2)( . (11.3)

rogorc vxedavT, (11.3)-is marcxena mxareSi dgas erTiani saSua-

losagan, yvela monacemis gaerTianebiT miRebuli SerCevis y saSualo-

sagan, gadaxrebis kvadratebis jami (gaixseneT SerCeviTi dispersiis ag-

ebis gza). am sidides mokled aRvniSnavT tss-iT (Total Sam of Squares).aseve, Tu davakvirdebiT (11.3)-is marjvena mxaris pirvel Sesakrebs, Si-

da jami warmoadgens i-uri jgufis saSualosagan gadaxrebis kvadrateb-is jams. amitom (11.3)-is marjvena mxaris pirvel Sesakrebs vuwodoT

gadaxra jgufis SigniT; Sesabamisad, is aRvniSnoT wss-iT (Within Samof Squares). bolos, (11.3)-is marjvena mxaris meore Sesakrebi, ramdena-

132

dac is warmoadgens jgufuri saSualoebis saerTo saSualosagan gada-xrebis kvadratebis jams, SeiZleba gaviazroT, rogorc jgufTaSorisgadaxris sazomi da aRvniSnoT is bss-iT (Between Sam of Squares). maSa-sadame, sabolood,

tss =

k

i

n

jij

i

yy1 1

2)( , wss =

k

i

n

jiij

i

yy1 1

2)( , bss =

k

i

n

ji

i

yy1 1

2)( (11.4)

da (11.3) SeiZleba mokled ase SeiZleba gadavweroT: tss = wss + bss.SevTanxmdeT, rom im SemTxveviT sidideebs, romelTa konkretul

realizaciebsac warmoadgens tss , wss , bss ricxvebi, Cven Sesabamisi di-di asoebiT aRvniSnavT, anu TSS, WSS, BSS warmoadgenen SemTxveviT si-

dideebs, romlebic miiReba (11.2) – (11.4) gamosaxulebebSi yvela yij-is

SecvliT Yij-ebiT.

moxerxebulia, gamoTvlebis gamartivebis mizniT, SemoviRoT Sem-degi aRniSvnebi:

ii

n

jiji ynyy

i

1

- monacemebis jami i-ur jgufSi; (11.5)

ynyyk

i

n

jij

i

1 1

- monacemebis jami yvela jgufSi. (11.6)

maSin advili saCvenebelia, rom

tss =n

yyyy

k

i

n

jij

k

i

n

jij

ii 2

1 1

2

1 1

2)(

. (11.7)

bss =

k

i i

ik

i

n

ji n

y

n

yyy

i

1

22

1 1

2)( . (11.8)

cxadia, rom am formulebis gamoyeneba kargia maSin, roca mocem-

ulia uSualod monacemebi. magram Tu monacemebi dajgufebulia saSua-

loebisa da dispersiebis mixedviT, rogorc es gakeTebulia 11.1 magal-

iTSi, maSin ukeTesia gamoviyenoT Semdegi formulebi:

bss =2

11

2 1

k

iii

k

iii yn

nyn . (11.9)

wss =

k

iii sn

1

2)1( . (11.10)

gamovTvaloT zemoT SemoRebuli sidideebis ricxviTi mniSvnel-

obebi 11.1 magaliTisaTvis:

bss = [2003.782+2003.32+503.322+2003.232+2002.732+2002.592]-

- (1/1050)[2003.78+2003.3+503.32+2003.23+2002.73+2002.59]2 = 184.38,

wss = 1990.792+1990.772+490.862+1990.782+1990.812+1990.822 = 663.87.

133

SemoviRoT kidev Semdegi sidideebi: bms = bss /(k -1) – jgufTaSo-risi kvadratebis saSualo (Between Mean Square) da wms = wss /(n - k) –jgufis SigniT kvadratebis saSualo (Within Mean Square).

intuiciurad, statistikuri kriteriumis asagebad ase vmsjel-

obT: Tu bms/wms Sefardeba “didia”, maSin jgufebi “sxvadasxva saSua-

lo yofaqcevis” individebiT yofila dakompleqtebuli da maSasadame,

H0 hipoTeza unda uarvyoT da piriqiT, Tu bms/wms Sefardeba “mci-

rea”, maSin jgufebi “erTnairi saSualo yofaqceviT” xasiaTdeba da ma-

Sasadame, H0 hipoTezis uaryofis safuZveli ar unda gvqondes.

statistikuri kriteriumi damyarebulia swored F = BMS / WMSSefardebaze, romelsac mtkicdeba, rom rogorc (11.1) modeliT agebulSemTxveviT sidides, aqvs F(k-1,n-k)-ganawileba.

sabolood, statistikuri kriteriumi ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis F statistikisdakvirvebuli f = bms / wms mniSvnelobisaTvis

f > Fk –1,n – k, 1- ,maSin H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi, amis safuZveli

ara gvaqvs. Sesabamisi p-mniSvneloba tolia

p = P{ Fk –1,n – k > f }. (11.11)

davubrundeT 11.1 magaliTs da vnaxoT, ra Sedegs mogvcems es

kriteriumi. rogorc vnaxeT, bss = 184.38, wss = 663.87, amitombms = bss /(k -1)=184.38/(6-1)=36.875,

wms = wss /(n - k) = 663.87/(1050-6) = 0.636da kriteriumis statistikis ricxviTi f mniSvneloba tolia:

f = bms / wms = 36.875/0.636 = 57.979.F(5,120)-ganawilebis cxrilebidan vpoulobT, F5, 120, 0.999 = 4.42.

amitom F5, 1044, 0.999 < F5, 120, 0.999 = 4.42 < 57.979 = f, saidanac davaskvniT,rom p = P{ F5 , 1044 > f }= P{ F5 , 120 > f }= P{ F5 , 120 > 57.979}< 0.001, anuSedegi statistikurad mniSvnelovania da maSasadame, H0 hipoTezas,

rom yvela jgufs aqvs erTnairi saSualo, = 0.001 mniSvnelovnebis

doniTac ki uarvyofT.

11.3. jgufTa Sedareba erTfaqtorian ANOVA modelSi.

wyvilTa Sedarebis t -kriteriumi.

wina nawilSi H0 : 1 = 2 =…= k = 0 hipoTezis uaryofis SemT-xvevaSi Cven vaskvniT, rom jgufebis saSualo yofaqceva ar aris erTna-

iri, magram kerZod romeli jgufebia gansxvavebuli, amas ver vambobT.

axla swored am sakiTxze gvinda visaubroT. H0 hipoTezis uaryofa ni-

Snavs imas, rom arsebobs sul cota ori jgufi, romelTa saSualoebic

134

mniSvnelovnad gansxvavebulia. davuSvaT, esenia m-uri da l-uri jgufe-

bi. (11.1) modelis mixedviT, yoveli i-saTvis

in

jij

ii Y

nY

1

1( i=1,2,…,k)

SemTxveviTi sidide normaluradaa ganawilebuli saSualoTi, + i

da dispersiiT, 2/ni. amitom gasagebia, rom

lm YY ~ N(m - l ; 2(1/nm +1/nl), (11.12)

xolo H0 hipoTezis samarTlianobis SemTxvevaSi,

lm YY ~ N(0 ; 2(1/nm +1/nl). (11.13)

(11.13) Tanafardobidan gamomdinare, cnobili -s SemTxvevaSi,

cxadia, kriteriumis statistikad aviRebdiT)/1/1(2

lm

lm

nn

YY

SemTx-

veviTi sidides, romelsac cxadia eqneba standartuli normaluri gana-

wileba. magram rogorc wesi, ucnobia. maSin rogor SevafasoT es si-

dide? amisaTvis gavixsenoT Tu rogor viqceodiT saSualoTa tolobis

t–kriteriumis agebisas oramokrefian amocanaSi toli, ucnobi dispers-

iebis SemTxvevaSi. maSin 2 dispersias vafasebdiT gaerTianebuli SerCe-vis safuZvelze Semdegnairad:

2

)1()1(

21

212

2112

,21

2

nn

SnSnS

nn

nna.

axla, roca laparakia k (da ara or) cal amokrefaze, moviqceT

analogiurad, anu SevkriboT jgufebis Tavisuflebis xarisxebiT Sewo-

nili dispersiebi da gavyoT Tavisuflebis xarisxebis jamze:

1 2

2

2 2 21 1 2 22 2

, ,...,1 2

( 1) ( 1) ... ( 1)

1 1 ... 1k

a k

n n k knn n n

k

n S n S n SS S

n n n

2

1 1

( ( 1) ) /( )i

k k

i in ii i

n S n k

, (11.14)

saidanac (11.10)-sa da WMS–is ganmartebiT sabolood davaskvniT, rom

S2 = WSS/(n-k) = WMS. (11.15)Sesabamisad, kriteriumis statistikas ucnobi dispersiis SemTx-

vevaSi eqneba Semdegi saxe:

)/1/1(,

lm

lmlm

nnWMS

YYT

, (11.16)

romelsac mtkicdeba, rom aqvs t(n-k)-ganawileba da H0 : m = l hipoTe-

zis Sesamowmebeli kriteriumi H1 : m l alternativis winaaRmdegase Camoyalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis Tm,l statistik-

is dakvirvebuli tm ,l mniSvneloba akmayofilebs Semdeg pirobas

tn-k, /2 tm ,l tn-k, 1-/2, (11.17)

135

maSin mniSvnelovnebis doniT H0 : m = l hipoTezis uaryofis safuZ-veli ara gvaqvs, winaaRmdeg SemTxvevaSi ki mas uarvyofT. Sesabamisi p-mniSvneloba ase gamoiTvleba:

p = 2 P{Tn-k < tm ,l }, tm ,l < 0 da p = 2 P{Tn-k > tm ,l }, tm ,l 0. (11.18)

davubrundeT 11.1 magaliTs da vnaxoT aris Tu ara statistiku-

rad mniSvnelovani gansxvaveba aramwevelebsa da Tundac pasiur mwevel-ebs Soris. amisaTvis gamovTvaloT T1,2 statistikis mniSvneloba: t1,2 == (3.78-3.3)/[0.636(1/200+1/200)]1/2 = 6.02 da p = 2 P{t1044 > t1,2 }< 0.001.Sesabamisad, gansxvaveba aramwevelebsa da pasiur mwevelebs Sorisac ki

statistikurad mniSvnelovania. analogiurad Sedardeba danarCeni wyvi-

lebic. yvela wyvilis Sedareba mocemulia Semdeg cxrilSi:

Sesadarebeliwyvilebi

statistikis ricxviTimniSvneloba

p-mniSvneloba,mniSvnelovneba

NS, PS t1,2 = 6.02 < 0.001NS, NI t1,3 = 3.65 < 0.001NS, LS t1,4 = 6.90 < 0.001NS, MS t1,5 = 13.17 < 0.001NS, HS t1,6 = 14.92 < 0.001PS, NI t2,3 = -0.16 umniSvneloa

PS, LS t2,4 = 0.88 umniSvneloa

PS, MS t2,5 = 7.15 < 0.001PS, HS t2,6 = 8.90 < 0.001NI, LS t3,4 = 0.71 umniSvneloa

NI, MS t3,5 = 4.68 < 0.001NI, HS t3,6 = 5.79 < 0.001LS, MS t4,5 = 6.27 < 0.001LS, HS t4,6 = 8.03 < 0.001MS, HS t5,6 = 1.76 umniSvneloa

SevniSnoT, rom (11.14)-Si Cven ucnobi dispersia 2 SevafaseTyvela jgufis monacenebis gamoyenebiT, da ara mxolod Sesadarebeli

jgufebis monacemebis saSualebiT, rogorc es 10.3 punqtSi xdeboda. es

gamowveulia imiT, rom 2 yvela jgufisaTvis Cveni modelis mixedviT,

erTidaigivea da 10.3-Si Catarebuli msjelobis msgavsad (SerCevis moc-

ulobis zrda iwvevs saSualos standartuli Secdomis Semcirebas)

yvela jgufis gaerTianebiT miRebuli Sefaseba ukeTesia, vidre mxol-od ori jgufiT. zogjer, roca yovel jgufs Tavisi (sxvadasxva) dis-

persia aqvs, maSin wyvilebis Sesadareblad unda gamoviyenoT 10.5 punq-

tSi SemoTavazebuli wyvilTa t-kriteriumi.is faqti, rom (11.16)-iT gansazRvrul SemTxveviT Tm , l sidides

aqvs t(n-k)-ganawileba, Cven SegviZlia gamoviyenoT ndobis intervalebis

136

asagebad jgufebis WeSmarit saSualoTa m-l sxvaobisaTvis. 1 ndo-bis albaTobis mqone anu (1 )100%-iani ndobis intervals aqvs Semd-

egi saxe:

lmknlm

lmknlm nn

StYYnn

StYY11

;11

2/1,2/1, , (11.19)

sadac rogorc (11.15)-dan gvaxsovs, S 2 = WSS/(n-k) = WMS.

11.4. wrfivi kontrastebi.

ganxilva daviwyoT 11.1 magaliTis Semdegi modifikaciiT.

magaliTi 11.2. davuSvaT, Cven gvinda filtvebis funqciurobis

Sedareba erTad aRebuli bolo sami jgufis mwevelebsa (am jgufis pi-

robiTi aRniSvnaa IS (inhale smokers)) da aramwevelebis jgufebs Soris,

Tu gamokvlevebiT dadgenilia, rom am jgufis mwevelebis 70% arisMS jgufis mweveli, 20% aris HS jgufis mweveli da 10% – LSjgufis mweveli. rogor SevadaroT IS tipis mwevelebi rogorc jgufi

aramwevelebis jgufs?

amisaTvis SemoviRoT e.w. wrfivi kontrastebis cneba. wrfivi ko-ntrasti (igi aRiniSneba L asoTi) ewodeba jgufTa saSualoebis nebis-

mier wrfiv kombinacias

k

iii YcL

1

, romlisaTvisac 01

k

iic . (11.20)

SevniSnoT, rom ci koeficientebis specialuri SerCevis xarjze

Cven SegviZlia miviRoT ukve Seswavlili saSualoTa sxvaobebi. marTl-

ac, Tu aviRebT cm = 1, cl = -1, xolo yvela danarCeni i-ebisaTvis ci = 0,

maSin cxadia, rom 01

k

iic da (11.20) mogvcems L = lm YY . Tu davubr-

undebiT 11.2 magaliTs, maSin is, rac Cven gvainteresebs aris Semdegi

tipis sxvaoba: L = 6541 2.07.01.0 YYYY , anu wrfivi kontrasteb-

is enaze rom vTqvaT, Cven gvainteresebs wrfivi kontrasti Semdegi ko-

eficientebiT: c1 = 1, c2 = c3 = 0, c4 = -0.1, c5 = -0.7, c6 = -0.2.yovelive zemoTqmulidan gamomdinare, romelime jgufebis (an ma-

Ti gaerTianebebis) Sedarebis amocana, sabolood daiyvaneba imis Semow-

mebaze, aris Tu ara ama Tu im wrfivi kontrastis L saSualo 0-istoli, anu Cveni amocanaa SevamowmoT H0 : L = 0 hipoTezis samarTlian-

oba H1 : L 0 alternativis winaaRmdeg. am amocanis gadasaWrelad Sev-

niSnoT, rom 01

k

iic pirobis gamo

k

iii

k

iii

k

ii

k

iii

k

iiiL ccccYEcEL

11111

)( , (11.21)

137

k

i i

ik

i ii

k

iii

k

iii n

c

ncYDcYcDDL

1

22

1

22

1

2

1

. (11.22)

cxadia, rom ucnobi 2 Sefasdeba (11.15) tolobiT gansazRvru-li S2 sididiT da kriteriumis statistikas eqneba Semdegi saxe:

k

i i

i

L

n

cS

LT

1

22

, (11.23)

romelsac mtkicdeba, rom aqvs t(n-k)–ganawileba da amitom statistiu-ri kriteriumi ase Camoyalibdeba:

Tu mocemuli mniSvnelovnebis donisaTvis TL statistikis da-

kvirvebuli tL mniSvneloba akmayofilebs pirobas

tn – k , /2 tL tn – k , 1- /2, (11.24)

maSin mniSvnelovnebis doniT H0 : L = 0 hipoTezis uaryofis safuZve-

li ara gvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. Sesabamisi p-mniS-vneloba ase gamoiTvleba:

p = 2 P{Tn-k < tL }, Tu tL < 0 da p = 2 P{Tn-k > tL }, Tu tL 0. (11.25)davubrundeT 11.2 magaliTs da gavixsenoT, rom s2 = 0.636. garda

amisa, L = 6541 2.07.01.0 YYYY wrfivi kontrastis SerCeviTi l

mniSvneloba tolia l = 3.78 – 0.1 3.23 – 0.7 2.73 – 0.2 2.59 = 1.03. amitom

2 2 2 2

1.0314.69

0.636 (1 / 200 ( 0.1) / 200 ( 0.7) / 200 ( 0.2) / 200)Lt

.

cxadia, rom p = 2 P{T1044 > 14.69}< 2 P{T120 > 14.69}< 2 0.0005 = 0.001,saidanac davaskvniT, rom = 0.001 mniSvnelovnebis doniT H0 : L = 0 hi-poTeza unda uarvyoT, anu IS jgufis mwevelebs gacilebiT dabali fi-

ltvebis funqciuroba aqvT, vidre aramwevelebs.

wrfivi kontrastebis gamoyeneba SeiZleba maSinac, roca gansxva-

vebul jgufebs dozebis gansxvavebuli doneebi SeesabamebaT da kontra-

stis koeficientebi SeirCeva kerZo dozebis Sesabamisad. es midgoma

mniSvnelovania maSin, roca SerCevaTa moculobebi araa didi da jgufe-

bis yvela wyvilis Sedareba ar iZleva mniSvnelovan gansxvavebas, magr-

am mTlianobaSi monacemebs aqvT mkveTrad gamokveTili mimarTulebis

trendi. ganvixiloT Semdegi

magaliTi 11.3. davuSvaT, Cven gvainteresebs axdens Tu ara mniSv-nelovan gavlenas dRis ganmavlobaSi moweuli sigaretis Rerebis rao-

denoba FEF–is sidideze IS jgufis mwevelebisaTvis.

amoxsna. rogorc gvaxsovs, susti mwevelebi dReSi 1-dan 10 Re-

ramde ewevian. CavTvaloT, rom saSualod isini (1+10)/2 = 5.5 Rers ewe-vian. aseve,Lzomierad mwevelebi dReSi 11-dan 39 Reramde, anu saSual-

od (11+39)/2 = 25 Rers ewevian da CavTvaloT, rom magari mwevelebi

138

zustad 40 Rers ewevian (da ara mets) dRis ganmavlobaSi. maSin Cveni

amocanaa L = 654 40255.5 YYY kontrastis statistikuri mniSvnel-

ovnebis Semowmeba. magram rogorc vxedavT, kontrastis koeficientebis

jami 5.5+25+40 = 70.5 da is araa 0-is toli. imisaTvis, rom es piroba

Segvisruldes, TiToeuli kontrastis koeficienti SevamciroT koefi-cientebis jamis mesamediT (radgan sam jgufzea laparaki), anu

70.5/3=23.5 sididiT da statistikuri mniSvnelovnebis Sesamowmeblad

ganvixiloT Semdegi wrfivi kontrasti

4 5 6 4 5 6(5.5 23.5) (25 23.5) (40 23.5) 18 1.5 16.5L Y Y Y Y Y Y .

wina amocanis msgavsad, s2 = 0.636, L wrfivi kontrastis SerCevi-

Ti l mniSvneloba tolia l = – 18 3.23 + 1.5 2.73 + 16.5 2.59 = -11.31. am-itom

19.838.1

31.11

)200/)5.16(200/)5.1(200/)18((636.0

31.11222

Lt .

cxadia, rom p = 2 P{T1044 < -8.19}< 2 P{T120 < -8.19}< 2 0.0005 = 0.001,saidanac davaskvniT, rom = 0.001 mniSvnelovnebis doniT H0 : L = 0 hi-poTeza unda uarvyoT, anu IS jgufis mwevelebSi gacilebiT dabali

filtvebis funqciuroba aqvT maT vinc mets ewevea.

11.5. mravlobiTi Sedareba (bonferonisa da Sefes meTodebi).

rogorc 11.3 punqtSi vnaxeT, dispersiuli analizis Catarebis

Semdeg H0 hipoTezis uaryofis SemTxvevaSi saWiro gaxda wyvilTa Sed-

areba, imisaTvis, rom gagvego kerZod romel jgufTa generaluri saSu-

aloebi iyo gansxvavebuli. rogorc gvaxsovs, k = 6-is SemTxvevaSi CvendagvWirda 15 aseTi wyvilis ganxilva. sazogadod, gasagebia, rom dagv-

Wirdeba k(k –1)/2 aseTi wyvilis ganxilva, rac k -s didi mniSvnelobe-

bisaTvis sakmaod Sromatevadi da mouxerxebeli saqmea. ra Tqma unda,

kargi iqneboda iseTi kriteriumis ageba, romelic jgufebis konkretu-

li wyvilisaTvis ki ar “daiWerda gansxvavebas” generalur saSualoebs

Soris, aramed, erTdroulad yvela wyvilisaTvis. aseTi kriteriumebi

arsebobs da maT mravlobiTi Sedarebis kriteriumebs eZaxian. am nawil-Si Cven SevexebiT or aseT kriteriums, bonferonisa da Sefes kriteri-

umebs. pirveli maTgani gamoiyeneba uSualod jgufTa wyvilebisaTvis,

xolo meore ki zogadad wrfivi kontrastebisaTvis.

11.5.1. bonferonis mravlobiTi Sedarebis meTodi.

es meTodi formalurad Zalian hgavs 11.3 punqtSi ganxilul

wyvilTa Sedarebis t-kriteriums, magram maT Soris aris erTi mniSvne-lovani gansxvaveba, romelsac kriteriumis Camoyalibebisas aRvniSnavT.

jer erTi, Cven kvlav gvinda H0 : m = l hipoTezis Semowmeba H1 : m

139

l alternativis winaaRmdeg, mxolod axla ukve ara konkretuli m da

l-saTvis, aramed yvela ml-saTvis. kriteriumis statistikas isev iseTisaxe aqvs, rogorc (11.16)-Si:

)/1/1(,

lm

lmlm

nnWMS

YYT

, (11.26)

romelsac isev aqvs t(n-k)-ganawileba. magram dasaxelebuli mniSvnel-ovnebis donisaTvis Tm,l statistikis dakvirvebuli tm,l mniSvneloba un-

da Sedardes t(n-k)-ganawilebis ara /2 da 1-/2 kvantilebs, aramed

Sesabamisad, */2 da 1- */2 kvantilebs, sadac

*= /(k(k-1)/2). (11.27)

marTlac, rogorc vTqviT, k jgufisaTvis Sesadarebeli wyvileb-

is raodenobaa k(k-1)/2; davuSvaT, yvela wyvili Sedarebulia * mniSv-

nelovnebis doniT. vTqvaT, E aris xdomiloba imisa, rom Sedareba erTi

mainc wyvilisaTvis statistikurad mniSvnelovania. Cven gvinda, rom

P{E}= . rogor avarCioT *? cxadia, rom Tu yoveli wyvilis Sedar-

ebiT miRebuli Sedegi damoukidebelia, maSin P{EC}= (1- *)k(k-1)/2, saida-

nac davaskvniT, rom * ise unda avarCioT, rom 1- = (1- *)k(k-1)/2. mag-

ram sakmarisad mcire *- saTvis (1- *)k(k-1)/2 1- (k(k-1)/2) * da ami-

tom 1- = 1- (k(k-1)/2) * gantolebis amoxsniT vRebulobT (11.27)-s.

Sesabamisad, bonferonis statistikuri kriteriumi ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis Tm,l statistik-is dakvirvebuli tm,l mniSvneloba akmayofilebs pirobas

2/, *knt

tm ,l 2/1, *knt , (11.28)

maSin mniSvnelovnebis doniT H0 : m = l hipoTezis uaryofis safuZ-

veli ara gvaqvs, winaaRmdeg SemTxvevaSi mas uarvyofT.

SevniSnoT, rom radgan *= /(k(k-1)/2) < ,2/, *kn

t

< tn-k, /2 da

2/1, *knt > tn-k, 1-/2 da amitom (11.28)-Si sazRvrebi (11.17)-Tan SedarebiT

ufro farTea. garda amisa, es sazRvrebi damokidebulia k-ze da k-szrdasTan erTad sazRvrebi farTovdeba. cxadia, rom es nakli mosalo-

dnelia, radgan axali sazRvrebi “pasuxs agebs” yvela m da l-saTvis,maSin, roca, (11.17)-iT ganmartebuli sazRvrebi “pasuxismgebelia” mxo-

lod konkrertuli m da l-saTvis.bonferonis meTodis gamoyenebis sailustraciod ganixiloT Sem-

degi

magaliTi 11.4. gamoiyeneT mravlobiTi Sedarebis bonferonis meT-

odi 11.1 magaliTis cxrilSi warmodgenili monacemebisaTvis da gaakeT-

eT Sesabamisi daskvnebi.

amoxsna. vTqvaT, = 0.05. maSin radgan jgufebis raodenoba eqv-sia, anu k = 6, amitom (11.27) formulis mixedviT, *= 0.05/((65)/2) =

140

=0.0033. isev n - k = 1044 da2/1, *kn

t = t1044, 0.99833 kvantilis sapovnelad

gamoviyenoT normaluri aproqsimacia (radgan 1044 sakmarisad didi ri-

cxvia), anu is faqti, rom t1044, 0.99833 z0.99833 = 2.935. gavixsenoT 11.3punqtSi mocemuli wyvilTa Sedarebis cxrili. yvela statistikurad

mniSvnelovan SemTxvevaTa Soris absoluturi mniSvnelobiT yvelaze pa-

tara SerCeviTi mniSvnelobaa 3.65 da isic ki metia 2.935-ze, rac imasniSnavs, rom isini bonferonis meTodis mixedviTac rCebian statistik-

urad mniSvnelovnad. ufro metic, is wyvilebi, romelTa Soris gansxv-avebac statistikurad umniSvneloa, bonferonis meTodis mixedviTac

arian statistikurad umniSvnelo. SevniSnoT, rom im cxrilis agebisas

rom gamogveyenebina = 0.05 mniSvnelovnebis done, maSin kritikuli

are iqneboda | t | > 1.96, maSin roca, rogorc vnaxeT, bonferonis meTod-is SemTxvevaSi, es area | t | > 2.935.

11.5.2. Sefes mravlobiTi Sedarebis meTodi.

am punqtSi Cveni amocanaa SevamowmoT H0 : L = 0 hipoTezis sama-

rTlianoba H1 : L 0 alternativis winaaRmdeg, mxolod yvelanairi

wrfivi kontrastebisaTvis. Cven isev gvaqvs k jgufi, moculobebiT ni

da n = n1 + n2 +…+ nk. kriteriumis statistikas isev is saxe aqvs, rac11.4 punqtSi, anu

k

i i

i

L

n

cS

LT

1

22

, (11.29)

romlis dakvirvebuli tL mniSvneloba (11.24)-gan gansxvavebiT, unda Se-

dardes (rogorc es Sefem aCvena) (11.30)-iT ganmartebul sazRvrebs,

xolo Sefes statistikuri kriteriumi Semdegnairad yalibdeba:

Tu mocemuli mniSvnelovnebis donisaTvis TL statistikis da-

kvirvebuli tL mniSvneloba akmayofilebs pirobas

- 1,,1)1( knkFk tL 1,,1)1( knkFk , (11.30)

maSin mniSvnelovnebis doniT H0 : L = 0 hipoTezis uaryofis safuZve-

li ara gvaqvs, winaaRmdeg SemTxvevaSi mas uarvyofT.

(11.29) da (11.30) Tanafardobebidan advilad igeba ndobis inte-rvali WeSmariti L kontrastisaTvis (iseve, rogorc es gakeTebulia

(11.19)-Si):

(l -2

21, ,1

1

( 1)k

ik n k

i i

cs k F

n

, l +2

21, ,1

1

( 1)k

ik n k

i i

cs k F

n

). (11.31)

Sefes meTodis gamoyenebis sailustraciod ganixiloT Semdegi

141

magaliTi 11.5. gamoiyeneT mravlobiTi Sedarebis Sefes meTodi11.1 magaliTis cxrilSi warmodgenili monacemebisaTvis da gaakeTeT

Sesabamisi daskvnebi.

amoxsna. rogorc 11.3 magaliTidan gvaxsovs, tL = -8.19. (11.30)-dan

gvaqvs 32.321.2555)1( 95.0,,595.0,1044,51,,1 FFFk knk .

radganac -8.19 < -3.32, amitom =0.05 mniSvnelovnebis doniT H0 : L == 0 hipoTezas uarvyofT.

11.6. hipoTezaTa Semowmeba erTfaqtorian ANOVA modelSi

SemTxveviTi efeqtebis dros.

rogorc 11.1 punqtSi aRvniSneT, erTfaqtorian ANOVA model-

Si, Yij = + i + Xij, i-uri jgufis maxasiaTebeli i faqtori (efeqti),SeiZleba iyos deterministuli an SemTxveviTi. 11.2 da 11.3 punqtebi

exeboda deterministuli i efeqtebis SemTxvevas. am nawilSi Cven Se-

viswavliT erTfaqtorian ANOVA models, romelSic i efeqtebi Sem-TxveviTi sidideebia, romlebic damoukidebelia Xij SemTxveviTi sidide-

ebisgan da romlebic ganawilebulia normalurad saSualoTi 0 da dis-persiiT 2

A . aseT models saxelad hqvia SemTxveviT efeqtebiani, erT-

faqtoriani ANOVA modeli. SevniSnoT, rom am modelSi saerTo sa-Sualos emateba (normaluri) SemTxveviTi sidide i da maSasadame,

sxvadasxva dakvirvebaSi gaCndeba i-is sxvadasxva SerCeviTi mniSvnelo-

ba. amitom dakvirvebad obieqtebs (individebs) Soris gansxvaveba damok-

idebulia am SemTxveviTi efeqtis gafantulobis xarisxze, anu i SemT-

xveviTi sididis 2A dispersiis mniSvnelobaze: Tu is didia, maSin in-

dividebs Soris gansxvaveba didia da Tu is mcirea, maSin individebi

faqtiurad “ar gansxvavdebian”. es efeqti swrafad SesamCnevi rom ga-xdes, gamovTvaloT axal modelSi Yij SemTxveviTi sididis dispersia:

DYij = D + Di + DXij = 2 + 2A . rogorc vxedavT, deterministul efeq-

tebiani modelisagan gansxvavebiT, dispersias gauCnda damatebiTi Sesak-

rebi 2A . statistikuri amocanac, yovelive zemoTqmulis gaTvaliswin-

ebiT, swored am Sesakrebis mimarT yalibdeba: nulovani hipoTezaa, rom

H0 : 2A = 0, anu SemTxveviTi efeqtebi umniSvneloa, xolo alternativa,

cxadia iqneba H1 : 2A > 0, anu SemTxveviTi efeqtebi mniSvnelovania. am

modelSi ganixilaven or SemTxvevas: roca yvela jgufis SerCeviTi mo-

culobebi tolia (e.w. balansirebuli SemTxveva) da roca isini gansxv-avebulia (Sesabamisad, arabalansirebuli SemTxveva).

advili saCvenebelia, rom ANOVA-s am modelSi

E(WMS) = 2. (11.32)aseve, balansirebul SemTxvevaSi,

142

E(BMS) = 2 + q 2A , (11.33)

sadac q n1 = n2 = … = nk , xolo arabalansirebul SemTxvevaSi,

E(BMS) = 2 + n0 2A , (11.34)

sadac

k

iinn

knn

1

220 )1(

1, (11.35)

xolo n n1 + n2 + … + nk. cxadia, rom Tu (11.35)-Si yvela ni-is aviR-

ebT q-s tols, maSin n0 = q (SeamowmeT!), sxva sityvebiT rom vTqvaT,

(11.34) warmoadgens (11.33)-is ganzogadebas. zogad SemTxvevaSi, mtkicd-

eba, rom n0< n n/k. Tumca, Cveulebriv, am sidideebs Soris gansxvavebamcirea. kriteriumis statistikad isev viyenebT F = BMS / WMS stati-

stikas, romelsac aqvs F(k-1,n-k)-ganawileba. rac Seexeba 2-sa da 2A -s,

maTTvis igeba Sefasebebi. am Sefasebebis asagebad, viyenebT (11.32),

(11.33) tolobebs balansirebul da (11.32), (11.34) tolobebs arabalan-

sirebul SemTxvevaSi. marTlac, Tu SevxedavT formulaTa am wyvilebs,

rogorc orucnobian gantolebaTa sistemebs 2-isa da 2A is mimarT,

advilad davrwmundebiT, rom 2 = E(WMS) da 2A = E(BMS-WMS)/q

balansirebul SemTxvevaSi da 2 = E(WMS) da 2A = E(BMS-WMS)/n0

arabalansirebul SemTxvevaSi. amitom orive SemTxvevaSi, 2 -is Sefaseb-

ad aviRebT 2~ = WMS -s da 2A -is Sefasebebad ki 2~

A = max((BMS-

WMS)/q , 0) da 2~A =max((BMS-WMS)/n0 , 0) sidideebs, Sesabamisad.

sabolood, statistikuri kriteriumi iseve yalibdeba, rogorc

deterministuli efeqtebis dros:

Tu dasaxelebuli mniSvnelovnebis donisaTvis F statistikisdakvirvebuli f = bms / wms mniSvnelobisaTvis

f > Fk –1,n – k, 1- ,maSin H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi amis safuZveli

ara gvaqvs. Sesabamisi p-mniSvneloba tolia

p = P{ Fk –1,n – k > f }. (11.36)

kriteriumis gamoyenebis sailustraciod gavarCioT Semdegi

magaliTi 11.6. cxrilSi mocemulia plazma estradiolis (plasmaestradiol) menopauzis Semdgomi logariTmebis mniSvnelobebi xuTi qalis

(gadiis) sisxlSi. eqsperimenti tardeboda Semdegnairad: yoveli sisx-

lis sinji daiyo or tol nawilad da aTive sinji gaigzavna analizis-

aTvis laboratoriaSi, ise, rom iq ar scodnodaT, Tu visi sisxlis an-

alizs akeTebdnen. miiRes Semdegi Sedegebi:

individi sinji 1 sinji 2 sxvaobis moduli saSualoebi

1 25.5 30.4 4.9 27.95

143

2 11.1 15.0 3.9 13.053 8.0 8.1 0.1 8.054 20.7 16.9 3.8 18.805 5.8 8.4 2.6 7.10

SeiZleba Tu ara individualuri da individebs Soris variaciebis sid-

ideebis Sefaseba am monacemebiT?

amoxsna. gamoTvlebis Sedegad miviRebT, bss = 2.65775, wss = 0.15,anu bms = 2.65775/(9-5) = 0.66444, wms = 0.15/5 = 0.03 da f = 0.66444/0.03= 22.15. F(4,5)-ganawilebis cxrilebidan vpoulobT, rom P{F4,5 > 22.15}< 0.0022, saidanac davaskvniT, rom dakvirvebadi sididis individualu-

ri saSualo mniSvnelobebs Soris gansxvaveba statistikurad mniSvnel-

ovania. Tu gavixsenebT 2~A -is 2~

A = max((BMS-WMS)/q , 0) Sefasebas, miv-

iRebT, rom 2~A = (0.6644-0.03)/2=0.317 da maS, individebs Soris variaci-

ebis sidide daaxloebiT 10-jer metia individualuri (individSi) var-iaciis wms = 0.03 sidideze.

amocanebi1. mkvlevars surs gamoscados maRali arteriuli wnevis dawev-

is sami gansxvavebuli meTodi. SemTxveviT SeirCa pacientTa sami jgufi.I jgufs aZlevdnen garkveul preparats, II jgufs utardeboda specia-

luri varjiSebi, xolo III jgufi icavda specialur dietas. oTxi kvi-

ris Semdeg TiToeul pacients gauzomes arteriuli wneva (monacemebi

moyvanilia qvemoT). 0.05 mniSvnelovnebis doniT SeamowmeT hipoTe-

za imis Sesaxeb, rom populaciaTa saSualoebs Soris ar arsebobs gan-

sxvaveba.

preparati varjiSi dieta

10 6 5

12 8 9

9 3 12

15 0 8

13 2 4

2. SemTxveviT SerCeul pacientTa sam jgufs utardeboda mkurna-

loba stresis donis dawevis sami gansxvavebuli meTodiT. specialuri

mowyobilobiT gazomil iqna (procentebSi gamosaxuli) TiToeuli pac-

ientis daweuli stresis done (monacemebi moyvanilia qvemoT). 0.05 mniSvnelovnebis doniT, SegviZlia Tu ara daaskvnaT, rom procentebis

saSualo am jgufebSi gansxvavebulia?

I meTodi II meTodi III meTodi

3 12 15

10 12 14

144

5 17 18

1 13 14

13 18 20

3 9 22

4 14 16

3. SemTxveviT SerCeuli aTletebi dayves sam jgufad da dauniS-

nes sami saxis dieta erTi Tvis ganmavlobaSi. erTi Tvis gasvlis Semd-

eg TiToeuli aTletis mier dietebis mixedviT daklebuli kilograme-

bi moyvanilia qvemoT. 0.05 mniSvnelovnebis doniT, SegviZlia Tu

ara daaskvnaT, rom dietebi gansxvavebulia?

dieta A dieta B dieta C3 10 8

6 12 3

7 11 2

4 14 5

8

6

4. jandacvis saministros ainteresebs vin daTanxmdeba nebayofi-

lobiT Tavis Tavze gamoscados arteriuli wnevis dasawevi axali pre-

parati. regionebis mixedviT miRebuli monacemebis qvemoTmoyvanili

cxrilis mixedviT, 0.01 mniSvnelovnebis doniT, SegviZlia Tu ara

daaskvnaT, rom TiToeul regionSi eqsperimentSi monawileobis msurve-

lTa raodenoba erTidaigivea?regioni I II III IV

Tanaxmaa 87 62 56 93

5. qvemoT moyvanilia nawlavis Cxirebis raodenoba sami sxvadas-

xva tbis garkveul farTobze 5 dRiani periodis ganmavlobaSi.

0.05 mniSvnelovnebis doniT aris Tu ara gansxvaveba Cxirebis rao-

denobis saSualoebSi?

kus tba lisis tba faravnis tba

45 97 33

53 82 35

41 99 3138 84 28

55 79 26

6. studentebi SemTxveviT gadaanawiles sam jgufSi da TiToeul

jgufs SesTavazes swavlebis gansxvavebuli meTodi. kursis damTavrebis

Semdeg Catarebuli gamocdis Sedegebi moyvanilia qvemoT. 0.05 mni-

Svnelovnebis doniT aris Tu ara mniSvnelovani gansxvaveba gamocdis

Sedegebis saSualoebSi jgufebis mixedviT?

I jgufi II jgufi III jgufi

145

87 82 9792 78 90

61 41 83

83 65 92

47 63 91

7. zoologiuri maRaziis mepatrones ainteresebs dakavSirebulia

Tu ara arCeuli cxovelis saxeoba adamianis sqesTan. qvemoTmoyvanili

monacemebis mixedviT, 0.1 mniSvnelovnebis doniT, SeamowmeT Sesaba-

misi hipoTeza.

cxovelis saxeoba

---------------------------------------------------- ZaRli kata Citi

mdedrobiTi 23 4 8

mamrobiTi 32 27 16

l e q c i a 11.

Tavi 12. oramokrafiani amocanebi binomuri proporci-

ebisaTvis. kategoruli monacemebi.

12.1. Sesavali.

wina leqciebSi Cven ganvixileT hipoTezaTa Semowmebis amocanebi

erTi, ori da meti SerCevisaTvis, sadac ZiriTadad daSvebuli iyo,

rom monacemebis Sesabamisi populaciebi normaluria, es ki, rogorc

viciT, uwyveti tipis SemTxveviTi sididea. aqamde araferi gviTqvamsdiskretul modelebSi hipoTezaTa Semowmebis sakiTxis Sesaxeb erTze

met amokrefian amocanebSi. Cven saqme gveqneba ara mxolod ricxviT,

aramed e.w. kategorul monacemebTanac, romlebic Tavisi bunebiT swor-

edac rom diskretulebi arian da ara uwyveti. rogorc ukve vnaxeT,

umniSvnelovaness diskretuli tipis SemTxveviT sidideebs Soris, war-

moadgens binomuri SemTxveviTi sidide. amitomac am leqciaSi Cven Sevi-

swavliT ori da meti binomuri populaciis Sedarebis sakiTxs. aseTi

amocanebis Seswavlis aucilebloba Cven SegviZlia davinaxoT uamrav

magaliTze. moviyvanoT erT-erTi maTgani.

magaliTi 12.1. samedicino wreebSi sakmaod gavrcelebulia azriimis Sesaxeb, rom mkerdis kibo qalebSi damokidebulia maTi pirveli

mSobiarobis asakze. kerZod, rac ufro gvian xdeba pirveli mSobiaro-

ba, miT ufro metia qalebSi mkerdis kibos gaCenis Sansi. amis dasadge-

nad Catarda saerTaSoriso gamokvleva, romelSic aSS-sTan erTad mona-

wileobda saberZneTi, iugoslavia, brazilia taivani da iaponia. qalebs

146

ekiTxebodnen maTi pirveli mSobiarobis asaks. aRmoCnda, rom 683 qals3220-dan, romelsac hqonda mkerdis kibo, pirveli mSobiaroba hqondaT

30-ze meti wlis asakSi, xolo im qalebs Soris, romlebsac es daavad-

eba ar hqoniaT, 1498 qalis pirveli mSobiarobis asaki iyo 30 welze

meti 10245 SemTxvevidan. aris Tu ara mniSvnelovani gansxvaveba am mon-

acemebs Soris?

rogorc magaliTidan vxedavT, dakvirvebadi sidideebi warmoad-

genen garkveuli xdomilobebis moxdenaTa raodenobebs da Sesabamisad,

isini ver iqnebian uwyveti SemTxveviTi sidideebis gaTamaSebis Sedegi.

gasagebia, rom maT ukan diskretuli SemTxveviTi sidideebi dgas. rog-

oria es SemTxveviTi sidideebi, rogori ganawileba SeiZleba maT hqon-des da rogor Camoyalibdeba da Semowmdeba hipoTezebi? ai sakiTxebis

is (SeiZleba arasruli) nusxa, romelzedac am leqci-aSi gveqneba lap-

araki.

12.2. ori binomuri proporciis Sedareba, normaluri

aproqsimacia.

ganvagrZoT 12.1 magaliTis ganxilva. albaTobebi imisa, rom SemT-

xveviT amorCeuli qalis asaki pirveli mSobiarobisas metia an toli

30 welze daavadebulTa da janmrTelTa Soris Sesabamisad, aRvniSnoTp1-iTa da p2-iT. gasagebia, rom maSin CvenTvis sainteresoa pasuxi Seki-

Txvaze, aris Tu ara p1 da p2 erTmaneTis toli? cxadia, rom is raod-

enobebi, romlebic aRwerilia 12.1 magaliT (683 da 1498), SeiZleba

sul sxva yofiliyo, Tu Cven gamovkiTxavdiT sxva 3220 da 10245qals. amitom Cven vuSvebT, rom arsebobs SemTxveviTi sidideebi X1 da

X2, romelTa konkretuli gaTamaSebis Sedegadac miviReT swored es

ricxvebi, anu x1 = 683 da x2 = 1498. rogoraa ganawilebuli X1 da X2

SemTxveviTi sidideebi? Tu Cven davuSvebT, rom yoveli qalis pirveli

mSobiarobis asakis moxvedra ama Tu im intervalSi danarCenebisagan

damoukidebelia (rac bunebrivia), maSin cxadia, rom X1 da X2 sididee-bi warmoadgens binomurad ganawilebul SemTxveviT sidideebs cdaTa

sxvadasxva ricxviT, 3220 da 10245, sazogadod, ki n1-iTa da n2-iT. ga-

rda amisa, bunebrivia, vifiqroT, rom X1 da X2 SemTxveviTi sidideebi

damoukidebelia. maSin nulovani da alternatiuli hipoTeza ase yalib-

deba: H0 : p1 = p2 da H1 : p1 p2. rogorc cnobilia, binomuri ganawile-

bis p parametris saukeTeso wertilovan Sefasebas warmoadgens fardo-

biTi sixSire, np~ = X / n: Cvens SemTxvevaSi,11

~np = X1 / n1 ,

22~

np = X2 / n2.

amitom Cveni kriteriumic swored am sidideebze unda avagoT, radgan

maTi siaxlove p1-isa da p2-is siaxlovis maCvenebelia. ganvixiloT

sxvaoba11

~np -

22~

np = X1 / n1 - X2 / n2, rogorc SemTxveviTi sidide da Seve-

147

cadoT davaxasiaToT is. sazogadod,11

~np -

22~

np SemTxveviTi sididis ga-

nawilebis gamoTvla sakmaod Sromatevadi da rTulia, magram roca n1

da n2 imdenad didia, rom n1 p1(1- p1) 5 da n2 p2(1- p2) 5, maSin, ro-gorc cnobilia, SesaZlebelia

11~

np da22

~np SemTxveviTi sidideebis no-

rmaluri aproqsimaciebi, anu SegviZlia CavTvaloT, rom11

~np ~N(p1;

p1(1- p1)/n1) da22

~np ~ N(p2; p2(1-p2)/n2). maSin

11~

np -22

~np sxvaoba warmoad-

gens damoukidebeli normaluri SemTxveviTi sidideebis sxvaobas, rome-

lic isev normaluradaa ganawilebuli da H0 : p1 = p2 hipoTezis samar-

Tlianobis SemTxvevaSi adgili aqvs Semdeg Tanafardobas:

11~

np -22

~np ~ N(0; p(1- p)(1/n1+ 1/n2)), (12.1)

sadac p p1 = p2. sabolood, rogorc es ukve araerTxel gavakeTeT, Tu

sxvaobas (12.1)-is marcxena mxareSi davanormirebT standartuli gadax-

riT, miviRebT, rom

(11

~np -

22~

np ) / )/1/1()1( 21 nnpp ~ N(0;1). (12.2)

samwuxarod, miRebul statistikas jer kidev ver gamoviyenebT

Cveni miznebisaTvis, radgan is Seicavs ucnob p parametrs. hipoTezis

mixedviT, orive SerCeva amokrefilia erTidaimave populaciidan da ami-

tom SesaZlebelia maTi gaerTianebiT miRebuli SerCevis safuZvelze

SevafasoT p parametri Semdegi sididiT

21

21, 21

~nn

XXp nn

. (12.3)

CavsvaT es sidide (12.2) gamosaxulebaSi p parametris magivradda ganvixiloT Semdegi statistika:

21 ,nnT (11

~np -

22~

np ) / )/1/1()~1(~21,, 2121

nnpp nnnn . (12.4)

mtkicdeba, rom21 ,nnT ~ N(0;1) da masasadame, am statistikis gam-

oyeneba SeiZleba statistikuri kriteriumis asagebad, romelic sabol-ood ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis21 ,nnT statist-

ikis dakvirvebuli21 ,nnt mniSvneloba akmayofilebs pirobas

z/2 21 ,nnt z1-/2, (12.5)

maSin mniSvnelovnebis doniT H0 : p1 = p2 hipoTezis uaryofis safuZv-

eli ara gvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. Sesabamisi p-mni-Svneloba gamoiTvleba formuliT

p = 2(1-(21 ,nnt )), Tu

21 ,nnt 0 da p = 2(21 ,nnt ), Tu

21 ,nnt < 0. (12.6)

davubrundeT 12.1 magaliT da pirvel rigSi SevamowmoT n1 p1(1-p1) 5 da n2 p2(1- p2) 5 pirobebis samarTlianoba

21 ,~

nnp -ebis dakvirveb-

uli ricxviTi mniSvnelobebisaTvis. gvaqvs:

148

21 ,~

nnp = (x1+ x2)/ (n1+ n2) = (683 + 1498)/(3220 + 10245) = 2181/13465 = 0.162.

amitom cxadia, rom

n221 ,

~nnp (1-

21 ,~

nnp ) n121 ,

~nnp (1-

21 ,~

nnp ) = 3220 0.1620.838 = 437.1 5.

gamovTvaloT axla

21 ,nnt = (683/3220-1498/10245)/( 0.1620.838(1/3220+1/10245))1/2 = 8.9.

vinaidan21 ,nnt = 8.9 > 0, amitom kriteriumis Sesabamisi p-mniSvne-

loba tolia p = 2(1-(21 ,nnt )) = 2(1-(8.9)) < 0.001 da maSasadame, H0 : p1

= p2 hipoTezas uarvyofT = 0.001 mniSvnelovnebis donisTvisac ki. e.i. pirveli mSobiaroba 30-ze meti wlis asakSi hqondaT mkerdis kiboTi

daavadebuli qalebis ufro maRal procents, (683 / 3220) 100% =21.2%, vidre am mxriv janmrTel qalebs, (1498/10245)100% = 14.6%.

12.3 ori binomuri proporciis Sedareba, SeuRlebis 22cxrili.

wina punqtSi ganxiluli magaliTi SegviZlia sxvanairadac amov-

xsnaT. cxadia, pasuxi igive darCeba, magram am gzas aqvs ganzogadebis

meti perspeqtiva. is gamodgeba ara mxolod ori binomuri proporciis,aramed ramdenime aseTi sididis Sesadarebladac. am tipis amocanebs mo-

kled proporciaTa erTgvarovnebis amocanebs eZaxian. garda amisa, ismeTodi, romlis ganxilvasac am punqtSi vapirebT, gamodgeba ori (an

meti) niSnis damoukideblobis Sesamowmebladac, razec momdevno punq-

tSi iqneba saubari. am meTods SeuRlebis 22 (sazogadod, rc) cxri-

lis meTodi hqvia. SeuRlebis 22 cxrili warmoadgens or striqonianda or svetian cxrils, romlis ujredebSic Cawerilia ori niSnis mi-

xedviT dayofili monacemebi (dakvirvebis Sedegebi). 12.1 magaliTisaT-

vis is ase gamoiyureba:

qalisAasaki pirveli

mSobiarobisas

cxrili 12.1

kategoria 30 < 30 sul

aqvs kibo 683 2537 3220

ara aqvs kibo 1498 8747 10245

sul 2181 11284 13465

rogorc vxedavT, ori striqonisa da ori svetis garda kidev

aris bolo striqoni da bolo sveti, romlsac marginaluri striqonida sveti ewodeba da romelSic warmodgenilia Sesabamisad, cxrilis

striqonebsa da svetebSi mdgomi ricxvebis jamebi. ra Tqma unda, Cven

149

SegviZlia aseTive cxrilis ageba zogad SemTxvevaSic, anu Semdegi sa-xis cxrilis ageba,

qalisAasaki pirveli

mSobiarobisas

cxrili 12.2


aqvs kibo x1 n1-x1 n1

ara aqvs

kibo

x2 n2-x2 n2

sul x1+x2 n1+ n2 –( x1+x2) n1+ n2

saidanac ufro naTlad Cans binomuri X1 da X2 SemTxveviTi sidideebis

n1 da n2 parametrebi da am sidideebis mier eqsperimentis Sedegad mi-

Rebuli x1 da x2 mniSvnelobebi. amitomac am cxrils dakvirvebuliSeuRlebis cxrilsac uwodeben. mis garda ageben e.w. mosalodnel Seu-Rlebis cxrilsac, romelSic dakvirvebuli x1 da x2 mniSvnelobebis na-cvlad Setanilia X1 da X2 SemTxveviTi sidideebis mosalodneli mniS-

vnelobebis Sefasebebi: n121 ,

~nnp = n1(x1+x2)/(n1+n2) da n2

21 ,~

nnp =

n2(x1+x2)/(n1+n2). mosalodnel SeuRlebis cxrils 12.1 magaliTisaTvis

aqvs Semdegi saxe:

qalisAasaki pirveli

mSobiarobisas

cxrili 12.3


aqvs kibo 521.6 2698.4 3220

ara aqvs kibo 1659.4 8585.6 10245

sul 2181 11284 13465

Sesabamisad, zogad SemTxvevaSi mosalodnel SeuRlebis cxrils

eqneba Semdegi saxe:

qalisAasaki pirveli

mSobiarobisas

cxrili 12.4


aqvs kibo n121 ,

~nnp n1(1-

21 ,~

nnp ) n1

ara aqvs kibo n221 ,

~nnp n2(1-

21 ,~

nnp ) n2

sul x1+x2 n1+ n2 –( x1+x2) n1+ n2

150

SevniSnoT, rom am cxrilebSi marginaluri striqonebi da svete-bi ar icvleba: cxrilis striqonebsa da svetebSi mdgomi ricxvebis

jamebi igivea, rac dakvirvebuli SeuRlebis cxrilebSi iyo. es kargad

Cans bolo cxrilidan:

n121 ,

~nnp + n1(1-

21 ,~

nnp ) = n1, n221 ,

~nnp + n2(1-

21 ,~

nnp ) = n2,

n121 ,

~nnp + n2

21 ,~

nnp = (n1+n2) 21 ,

~nnp = (n1+n2) (x1+x2)/(n1+n2) = x1+x2,

n1(1-21 ,

~nnp )+n2(1-

21 ,~

nnp )=(n1+n2)(1-21 ,

~nnp )=

=(n1+n2)(n1+n2 – (x1+x2))/(n1+n2) = n1+n2–( x1+x2).

SeuRlebis 22 cxrilis meTodis arsi imaSi mdgomareobs, rom

erTmaneTs SevadaroT 12.1 da 12.3 cxrilebis ujredebSi moTavsebuli

ricxvebi (sazogadod, 12.2 da 12.4 cxrilebis ujredebSi moTavsebuli

sidideebi): Tu erT ujredSi mainc miviReT “didi sxvaoba”, maSin hip-

oTeza p1 da p2 proporciebis tolobis Sesaxeb unda uarvyoT, xolo

winaaRmdeg SemTxvevaSi unda vaRiaroT, rom monacemebi ar gvaZlevs amhipoTezis uaryofis safuZvels. magram cxadia, aqve ismis kiTxva: ras

niSnavs “didi sxvaoba”? mtkicdeba, rom Tu Cven yovel ujredSi ganvi-

xilavT (O-E)2/E tipis sidides, sadac O Seesabameba dakvirvebuli (Obse-rved) SeuRlebis cxrilis ujredSi, xolo E ki mosalodneli (Expected)SeuRlebis cxrilis ujredSi mdgom sidideebs da miRebul oTxive si-

dides SevkrebT, maSin jams, rogorc SemTxveviT sidides aqvs 2(1)-gana-wileba da maSin, pasuxi SekiTxvaze, ras niSnavs “didi sxvaoba” naTe-

lia: miRebuli jamis sidide dasaxelebuli mniSvnelovnebis donisa-

Tvis unda Sedardes 2(1)-ganawilebis zeda 21,1 kvantilis ricxviT

mniSvnelobas; Tu jamis mniSvneloba aRmoCndeba 21,1 -ze didi, maSin

hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi, amis safuZveli ara

gvaqvs.

imisaTvis, rom mivceT statistikas formulis saxe, SemoviRoTaRniSvnebi, romlebic momavalSic dagvWirdeba: aRvniSnoT Oij-Ti dakvi-

rvebuli SeuRlebis cxrilis (i,j) ujredSi, xolo Eij-Ti ki mosalodn-

eli SeuRlebis cxrilis (i,j) ujredSi mdgomi sidideebi, i, j = 1,2. maS-in kriteriumis statistikas aqvs Semdegi saxe:

T = (O11-E11)2/E11 + (O12-E12)

2/E12 + (O21-E21)2/E21 + (O22-E22)

2/E22. (12.7)

radgan am statistikis ganawilebis dadgenisas xdeba diskretu-

li SemTxveviTi sidideebis aproqsimacia uwyvetiT, iatisma (Yates) Sem-oiRo misi Sesworeba, romelic ufro zustad uaxlovdeba 2(1)-ganawi-lebas, kerZod:

TYa = (|O11-E11| -1/2)2/E11 + (|O12-E12| -1/2)2/E12 + +(|O21-E21| -1/2)2/E21 + (|O22-E22| -1/2)2/E22 . (12.8)

151

am statistikis terminebSi kriteriumi ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis TYa statistik-

is dakvirvebuli tYa mniSvneloba akmayofilebs pirobas

tYa >21,1 , (12.9)

maSin mniSvnelovnebis doniT H0 : p1 = p2 hipoTezas uarvyofT, winaa-Rmdeg SemTxvevaSi, amis safuZveli ara gvaqvs.

davubrundeT magaliTs da gamovTvaloT tYa-is mniSvneloba:

tYa = (|683-521.6| -1/2)2/521.6 + (|2537-2698.4| -1/2)2/2698.4 +

+(|1498-1659.4| -1/2)2/1659.4 + (|8747-8585.6| -1/2)2/8585.6 = 77.89.

radgan 2999.0,1 = 10.83 < 77.89 = tYa, amitom H0 : p1 = p2 hipoTezas uarvy-

ofT, = 0.001 mniSvnelovnebis doniT da maSasadame, mkerdis kibo qale-

bSi mniSvnelovnadaa dakavSirebuli qalis pirveli mSobiarobis asakTan.

12.4. SeuRlebis 22 cxrili. niSanTa damoukidebloba.

rogorc aRniSnuli iyo, SeuRlebis 22 cxrili SeiZleba gamo-viyenoT ori niSnis damoukideblobis Sesadareblad. aseT amocanebSi

laparakia ara or (an met) populaciaze, aramed, erTidaimave populac-

iis ori sxvadasxva niSnis Sedarebaze. kerZod, Cven gvainteresebs axde-

nen Tu ara gavlenas isini erTmaneTze da maSasadame, arian Tu ara es

niSnebi albaTuri azriT damoukideblebi. sailustraciod moviyvanoT

beitsonis magaliTi:magaliTi 12.2. cxrilSi mocemulia tkbili muxudos yvavilis

ganawileba ori niSnis, yvavilis Seferilobisa da misi mtvrianas for-

mis mixedviT. Cveni amocanaa SevamowmoT am ori niSnis damoukideblob-

is hipoTeza cxrilSi moyvanili monacemebis mixedviT:

yvavilisASeferiloba cxrili 12.5

mtvrianas forma iisferi wiTeli sul

mogrZo 296 27 323

mrgvali 19 85 104

sul 315 112 427

sanam magaliTis amoxsnaze gadavidodeT, SevniSnoT, rom am cxri-

lSi marginaluri striqoni da sveti fiqsirebulia 12.1 cxrilisagan

gansxvavebiT, sadac fiqsirebuli mxolod marginaluri sveti iyo. ami-

tom aq SemTxveviTi faqtiurad, cxrilis mxolod erT-erTi ujredia

(magaliTad, (1,1) ujredi), xolo yvela danarCen ujredSi Casaweriricxvis mniSvneloba ganimarteba marginaluri mniSvnelobebiTa da am

152

ujredSi Cawerili sididis saSualebiT. miuxedavad am mniSvnelovanigansxvavebebisa, imave wesiT agebul TYa SemTxveviT sidides damoukideb-

lobis hipoTezis samarTlianobis SemTxvevaSi, aqvs isev 2(1)-ganawile-

ba da maSasadame, statistikuri kriteriumic igive rCeba. aRsaniSnavia,

rom gamoTvlebis TvalsazrisiT gacilebiT martivia am statistikis

ricxviTi mniSvnelobis gamoTvla Semdegi formulis saSualebiT:

TYa = n(|O11O22 - O12O21| -n/2)2/((O11+O12)(O11+O21)(O22+O12)(O22+O21)) (sadac n = O11 + O12 + O21 + O22). (12.10)

rogorc vxedavT, (12.10) formulaSi saerTod gaqra e.w. mosal-

odneli mniSvnelobebi, rac cxadia, gacilebiT amartivebs gamoTvlebs.

garda amisa, SevniSnoT, rom mniSvnelSi zis yvela marginaluri mniSvn-

elobebis namravli. gadavideT magaliTis amoxsnaze.

amoxsna. gamovTvaloT TYa statistikis dakvirvebuli tYa mniSvne-

loba:

tYa = 427(|29685-2719| - 427/2)2/315112323104 = 215.114.

vinaidan 2999.0,1 =10.83<215.114=tYa, amitom = 0.001 mniSvnelov-

nebis doniT H0 hipoTezas uarvyofT da maSasadame, yvavilis Seferilo-

ba da misi mtvrianas forma ar arian damoukidebeli niSnebi.

SevniSnoT, rom am magaliTSi Cven gamoviyeneT “warmatebis p al-

baTobis” saerTo21 ,

~nnp Sefaseba, rac SeiZleba sadao iyos. kerZod,

cnobilia, rom iisferi da mtvrianas mogrZo forma warmoadgenen domi-

nantur niSnebs, anu mendelis kanonis Tanaxmad, maTi Sexvedris albaT-

obebia 3/4, xolo misi wiTeli feri da mtvrianas mrgvali forma rec-

esiuli niSnebia da maTi Sexvedris albaTobebia 1/4. Tu es ori niSanimemkvidreobiT erTmaneTisagan damoukideblad gadaecema, maSin niSanTawyvilebis albaTobebi proporciuli unda iyos ricxvebis 9:3:3:1. maSa-sadame, damoukideblobis SemTxvevaSi 12.5 cxrili (dakvirvebuli SeuR-

lebis cxrili) unda Sedardes mosalodneli SeuRlebis Semdeg

cxrils:

yvavilisASeferiloba cxrili 12.6

mtvrianas forma iisferi wiTeli

mogrZo 4279/16 240.2 4273/16 80.05

mrgvali 4273/16 80.05 4271/16 26.7

Tu SevadarebT 12.5 da 12.6 cxrilebs igive (12.8) formuliT

gamoTvlili statistikaze dayrdnobiT, miviRebT

tYa = (|296-240.2| -1/2)2/240.2 + (|27-80.05| -1/2)2/80.05 +

+(|19-80.05| -1/2)2/80.05 + (|85-26.7| -1/2)2/26.7 = 218.22.

153

kvlav, radgan 2999.0,1 = 10.83 < 218.22 = tYa, amitom H0 hipoTezas

uarvyofT, = 0.001 mniSvnelovnebis doniT da vakeTebT igive daskvnas,

rom yvavilis Seferiloba da misi mtvrianas forma ar arian damoukid-

ebeli niSnebi.

amocanebi1. SeamowmeT 11.3 punqtSi gverdze warmodgenili cxrilis samar-

Tlianoba.2. cxrilSi mocemulia proteinis miRebis sidideebi qalebSi,

romlebic icaven sami tipis dietas, menopauzis Semdgom periodSi.

dietis tipi saSualo stand. gadaxra moculoba

STD 75 9 10LAC 57 13 10VEG 47 17 6

2.1. SeadareT jgufebis saSualoebi kritikuli mniSvnelobebiT;

2.2. gaakeTeT igive p-mniSvnelobaze dayrdnobiT;

2.3. SeadareT saSualoebi jufTa yoveli wyvilisaTvis t-kriteriumiT;2.4. davuSvaT, cnobilia, rom sazogadod, qalebis 70% icavs LAC ti-pis, xolo 30% ki VEG tipis dietas. gansazRvreT aris, Tu ara mniS-

vnelovani gansxvaveba Sesabamis wrfiv kontrastsa da 0-s Soris;2.5. mravlobiTi Sedarebis meTodiT gansazRvreT romeli populaciebis

saSualoebia gansxvavebuli.

3. Catarda gamokvleva imis dasdgenad, Tu romeli meTodiT jo-

bia sisxlis wnevis gansazRvra, standartuliT, Tu axliT. gazomvebi

aRiricxa 4 sxvadasxva adgilze da miiRes Semdegi Sedegebi:

gazomvis axali meTodi standartuli meTodi

adgi-

li

saSualo gadaxra mocul saSualo gadaxra mocul

A 142.5 21.0 98 142.0 18.1 98B 134.1 22.5 84 133.6 23.2 84C 147.9 20.3 98 133.9 18.3 98D 135.4 16.7 62 128.5 19.0 62

3.1. romeli midgoma ufro miesadageba am amocanas, determinituli, TuSemTxveviTi efeqtebis?

3.2. SeamowmeT aris Tu ara mniSvnelovani gansxvaveba oTxivegan.

4. SeadareT erTmaneTs inkubatorSi sxvadasxva temperaturis

dros gamoCekili wiwilebis masebi.

I – 80 82 81 82 73 85 81 83 80 82

II – 91 84 83 90 83 87 89 85 82 84

III – 81 83 84 87 76 84 88 83 91 84

154

1.1. gamoTvaleT sami saSualo;1.2. gansxvavdeba Tu ara mniSnelovnad ori saSualo mainc erTmaneTisa-

gan?

5. cxrilSi moyvanilia oTx sxvadasxva mdinareSi daWerili er-

Ti da igive jiSis Tevzis sigrZe.

1 265 315 360 290 420 310 290 340 280 270 255 290 300

325 280

2 305 340 310 320 270 300

3 300 380 380 350 335 335 370 280 320 405 300 380

4 280 260 295 280 270 305 350

2.1. rogor unda SeamowmoT hipoTeza, rom oTxive mdinareSi Tevzis ga-zrdisaTvis aris erTi da igive piroba;

2.2. SeamowmeT es hipoTeza 0.05 mniSvnelovnobis doniT.

5. cxrilSi moyvanilia tyis 4 adgilas SemTxveviT aRebul mci-

re ubnebze xavsiT dafaruli are (kv.sm.). pirveli ori adgili qaria-

nia, mesame da meoTxe qarisgan daculia.

adgili are

1 7 2 4

2 6 4 6

3 8 4 5

4 7 4 2 53.1. CamoayalibeT ori statistikuri hipoTeza;

3.2. airCieT kriteriumis statistika;

3.3. SearCieT mniSvnelovnobis done;

3.4. miiReT gadawyvetileba hipoTezis Sesaxeb.

6. CamoTvaleT daSvebebi, romlebic safuZvlad udevs erTfaqto-

riani dispersiuli analizis gamoyenebas.

7. ra pirobebSi eqneba populaciis monacemebis normaluri ganaw-

ilebidan gadaxras minimaluri gavlena pirveli gvaris Secdomaze erT-

faqtoriani dispersiuli analizisaTvis?

8. ganwyobilebaze damokidebuli mexsierebis fenomeni niSnavs,rom gaxseneba umjobesdeba, roca subieqts masalis gaxsenebisas igive

ganwyobileba aqvs, rac misi Seswavlis dros.

jgufis ganwyobileba

1 2 3 4 5

swavlis dros mxiaruli sevdiani neitraluri mxiaruli sevdiani

ganwyobileba

gaxsenebis dros

Seqmnili mxiaruli sevdiani neitraluri mxiaruli sevdiani

ganwyobileba

155

ganwyobilebaze damokidebuli mexsierebis Teoria winaswarmetyvelebs,rom jgufebi, romelTac aqvT swavlisa da gaxsenebis kongruentuli

mdgomareoba (e.i. jgufebi 1, 2 da 3) unda ixsenebdnen ufro mets, vid-

re jgufebi, romelTac hqondaT gansxvavebuli ganwyobilebebi swavlisa

da gaxsenebis dros (e.i. jgufebi 4 da 5). gaxsenebuli punqtebis rao-

denoba naCvenebia cxrilSi.

Gganwyob.

jgufi

1 8 12 12 7 10 9 13

2 14 13 6 9 12 10 8

3 9 12 7 7 11 6 10

4 5 9 6 8 4 6 10

5 6 9 8 8 5 4 5

romeli jgufis saSualoebi gansxvavdebian mniSnelovnad erTmaneTisagan

0.05 doneze?

9. eqsperimetSi monawileobda 3 jgufi. sakontrolo jgufs as-

mevdnen forToxlis wvens, placebo sakontrolo jgufs asmevdnen fo-

rToxlis wvens, magram eubnebodnen, rom masSi alkoholi iyo Sereuli,

mesame jgufs asmevdnen forToxlis wvens, masSi gaxsnili 0.8 ml. alk-

oholiT erT kg. wonaze. cdis pirebs misces sxvadasxva sigrZis monakv-

eTebi da sTxoves SeafasebinaT isini. TiToeuli afasebda 75 monakveTis

sigrZes. Ppirveli Sefasebis Semdeg cdis pirebs ebdnebodnen, rom maTi

Sefaseba ara sworia da meore Sefasebas sTxovdnen. cxrilSi moyvani-lia Secvlili Sefasebebis raodenoba.

jgufi

sakontrolo 18 2 11 3 26

placebo 17 19 26 4 18

alkoholi 31 27 16 24 41

cdis pirebi, romelTac miawodes alkoholuri sasmeli ufro xSirad

cvlian TavianT Sefasebebs, vidre cdis pirebi, romelTac miawodeswveni, Tu ara?

10. fsiqologebma ikvlies Zilis Taviseburebebi me-2, me-4 da me-

6 klasel bavSvebSi. TiToeul jgufSi iyo 14 bavSvi. Zilis periodi

ganisazRvreboda, rogorc drois raodenoba daZinebidan diliT gaRviZe-

bamde. cxrilSi moyvanilia Zilis periodebi wuTebSi.

Zilis periodi580 525 562 590 575 603 594 521 536 612 514 544 573 592

497 515 543 478 567 532 517 510 534 556 511 523 572 574

525 506 475 493 537 532 480 501 472 463 521 517 477 515

156

10.1. ipoveT TiToeuli jgufisaTvis saSualo da standartuli gadax-ra;

10.2. CamoayalibeT statistikuri hipoTezebi;

10.3. ra SegiZliaT daaskvnaT am qulebis safuZvelze dispersiuli ana-

lizidan?

10.4. saWiroa Tu ara mravaljeradi Sedarebis kriteriumebi? pasuxi

daasabuTeT.

11. beWdvis Secdomebis saSualo raodenoba iyo 18.5 im jgufisa-

Tvis, romelic usmenda klasikur musikas, 29.9 im jgufisaTvis, rome-

lic usmenda mZime roks da 17.5 im jgufisaTvis, romelic saerTod ar

usmenda musikas.11.1. sakmarisi Tu ara es statistikebi dispersiuli analizis Casatar-

eblad?

11.2. SegviZlia Tu ara davinaxoT musikis pirobebi mniSvnelovnad gans-

xvavdeba erTmaneTisagan Tu ara?

12. mdinaris napirze aSenebulia qarxana, saidanac gamoyenebul

wyals uSveben mdinareSi. cxrilSi moyvanilia eTridaigive dros mdi-

naris sxvadasxva adgilas aRebuli sinjis temperatura.

adgili temperatura saSualo

5km. zeviT 7 7 15 11 9 7.8

qarxanasTan 19 25 22 19 23 21.6

5km. qveviT 14 18 18 19 19 17.6

10km. qveviT 12 17 12 18 18 15.4

25km. qveviT 11 7 10 11 15 10.8

12.1. CamoayalibeT hipoTezebi;

12.2. dispersiuli analizis gamoyenebiT SeamowmeT Tqveni hipoTeza.

l e q c i a 12.

12.5. kategoruli monacemebis efeqtebis sazomebi.

wina punqtebSi Cven SeviswavleT hipoTezaTa Semowmebis amocane-

bi SeuRlebis cxrilebis saSualebiT, sadac yvelgan gamoiyeneboda 2-

kriteriumi. arsebobs kategoruli monacemebis daxasiaTebis sxva saSua-

lebebic (sazomebi), romlebzec am punqtSi gvinda visaubroT. daviwyoT

imiT, rom CavweroT SeuRlebis 22 cxrili e.w. safrTxe – avadmyofo-bis urTierTmimarTebis terminebSi, romelic xSirad Cndeba prospeqtu-

157

li, anu mosalodneli avadmyofobebis analizis dros. SeuRlebis 22cxrils aqvs Semdegi saxe:

avadmyofoba cxrili 12.7

safrTxe ki ara

ki a b a + b = n1

ara c d c + d = n2

a + c = m1 b + d = m2

sailustraciod moviyvanoT Semdegi

magaliTi 12.3. samedicino wreebSi aris eWvi imaze, rom OC (OralContraceptive) kontraceptivis miReba moqmedebs qalebSi gulis daavade-

baze. 3 wlis dakvirvebis Semdeg aRmoCnda, rom 40-44 ww. asakobriviintervalis qalebSi OC kontraceptivis 5000 momxmareblidan 13-sganuviTarda mikro-infarqti, xolo 10000 aramomxmarebels Soris mik-

ro-infarqti ganuviTarda mxolod 7-s. safuZvliania Tu ara eqimebis

eWvi?

ra Tqma unda am kiTxvaze pasuxis gacema Cven ukve SegviZlia wi-

na punqtebSi Seswavlili masalis gamoyenebiT, magram aq Cven gvinda

yuradReba gavamaxviloT sxva sidideebze, romelic SeiZleba dagvexmar-

os dasmul SekiTxvaze pasuxis gacemaSi. amitom Cven gavagrZelebT zo-

gad mimoxilvas da saWiroebis SemTxvevaSi davubrundebiT magaliTs.

aRvniSnoT kvlav p1-Ta da p2-iT Sesabamisad avadmyofobis ganviTa-

rebis albaTobebi safrTxis qveS myofTaTvis (magaliTSi OC -s momxma-reblTaTvis) da maTTvis, visac safrTxe ar emuqreba (magaliTSi OC -s

aramomxmareblTaTvis).

p1 da p2-is sxvaobas p1 - p2 riskebis sxvaobas uwodeben, xolo

maT Sefardebas RR p1 / p2 – fardobiT risks.rogorc cnobilia, p1 da p2-is wertilovani Sefasebebia

11~

np da

22~

np da Sesabamisad, p1 - p2 riskebis sxvaobis karg wertilovan Sefase-

bas warmoadgens11

~np -

22~

np SemTxveviTi sidide, romlis ganawilebac ap-

roqsimirdeba normaluri ganawilebiT, parametrebiT

p1 - p2 da p1(1- p1)/n1+ p2(1- p2)/n2,

roca n1 da n2 iseTia, rom n1 p1(1- p1) 5 da n2 p2(1- p2) 5, anu

11~

np -22

~np ~ N(p1 - p2 ; p1(1- p1)/n1+ p2(1- p2)/n2). (12.11)

amitom cxadia, Cven SegviZlia dispersiis gamosaxulebaSi ucno-

bi p1 da p2-is adgilas CavsvaT isev Sesabamisi Sefasebebi. gasagebia, rom

21

2211

212

22

1

112/121

~~)~1(~)~1(~

nnnnnn

ppn

pp

n

ppzppP

158

2

22

1

112/121

)~1(~)~1(~2211

n

pp

n

ppzpp

nnnn 1 - ,

saidanac davaskvniT, rom 100(1 - )%-iani ndobis intervals riskebis

sxvaobisaTvis aqvs Semdegi saxe: c1 p1 - p2 c2, sadac

2

22

1

112/1211

)~1(~)~1(~~~ 2211

21 n

pp

n

ppzppc

nnnnnn

, (12.12)

2

22

1

112/1212

)~1(~)~1(~~~ 2211

21 n

pp

n

ppzppc

nnnnnn

. (12.13)

cxadia, rom es Sefasebebi gamodgeba maSin, roca

n111

~np (1-

11~

np ) 5 da n2 22

~np (1-

22~

np ) 5.

davubrundeT magaliTs da avagoT 95%-iani ndobis intervali

riskebis sxvaobisaTvis. moxerxebulobisaTvis avagoT SeuRlebis 22cxrili:

mikro-infarqti cxrili 12.8

ki ara

xmarobs OC-s 13 4987 5000 = n1

ar xmarobs OC-s 7 9993 10000 = n2

20 = m1 14980 = m2

cxrilidan vaskvniT, rom11

~np = 13/5000 = 0.0026,

22~

np = 7/10000 = 0.0007

da11

~np -

22~

np = 0.0026 – 0.0007 = 0.0019. garda amisa, n111

~np (1-

11~

np ) =

=12.99 5 da n2 22

~np (1-

22~

np ) = 6.99 5 dabolos, (12.12) da (12.13)-dan

vpoulobT ndobis intervalis Semdeg sazRvrebs:

c1 = 0.0019 – 1.96(0.00260.9974/5000 + 0.00070.9993/10000)1/2 0.0004,c2 = 0.0019 + 1.96(0.00260.9974/5000 + 0.00070.9993/10000)1/2 0.0034.

maSasadame, sabolood davaskvniT, rom 95% SemTxvevebSi ucnobi

riskis sxvaobisaTvis gvaqvs 0.0004 p1 - p2 0.0034. SeiZleba Tu ara

aqedan saTanado daskvnis gakeTeba magaliTSi dasmul SekiTxvaze? ra

Tqma unda SeiZleba: radgan 95%-iani ndobis [0.0004, 0.0034] intervali

ar Seicavs 0-s, amitom 95% SemTxvevebSi gamoricxulia p1-isa da p2-is

toloba, anu eqimebis eWvi marTlac safuZvliania.

ra SeiZleba iTqvas RR = p1/p2 sidideze? misi wertilovani Sefa-

seba cxadia, tolia RR~

=11

~np /

22~

np . intervaluri Sefasebis asagebad,

Cven kvlav mogviwevs normaluri aproqsimaciis SesaZleblobis daSveba.

am SemTxvevaSi, naCvenebia, rom )~

ln( RR asimptoturad normaluria da

159

misi standartuli Secdoma tolia 21 // cndanb da amitom ndobis

100(1 - )%-ian intervlas RR -saTvis aqvs Semdegi saxe:

[ )~

ln( RR - z1- /2 21 // cndanb ; )~

ln( RR + z1- /2 21 // cndanb ]. (12.14)

ra Tqma unda, ndobis intervals sakuTriv RR -saTvis advilad

aRvadgenT (12.14)-dan da mas Semdegi saxe eqneba:

[exp( )~

ln( RR -z1- /2 )/()/( 21 cndanb ), exp( )~

ln( RR +

+ z1- /2 )/()/( 21 cndanb )]. (12.15)

Cveni magaliTisaTvis, )~

ln( RR = ln(11

~np /

22~

np ) = ln(0.0026/0.0007) = 1.312.

amitom (12.15) Tanafardobidan ndobis intervali iqneba

[exp(1.312 – 1.96(4987/(135000))), exp(1.312 + 1.96(4987/(135000)))] ==[1.5, 9.3].

garda riskebis fardobisa Semodis agreTve e.w. avadmyofobis Sa-nsebis fardoba, anu sidide

OR (p1/(1- p1))/(p2/(1- p2)) = p1(1- p2)/(p2(1- p1)). (12.16)

cxadia, rom mis wertilovan Sefasebas aqvs Semdegi saxe:

RO~

=11

~np (1-

22~

np ) /(22

~np (1-

11~

np )) = (ad)/(bc). (12.17)

Cveni magaliTisaTvis, RO~

= (139993)/ (74987) = 3.72. aqedan im daskvnisgakeTeba SeiZleba, rom OC-s momxmareblebs 3.72-jer ufro didi SansiaqvT mikro-infarqtis ganviTarebisa, vidre mis aramomxmareblebs.

ndobis intervalis asagebad Sansebis fardobisaTvis iyeneben

vulfis meTods, romlis mixedviTac, 100(1-)%-ian ndobis intervals

ln( RO~

)-saTvis aqvs Semdegi saxe saxe:

[ )~

ln( RO - z1- /2 dcba /1/1/1/1 , )~

ln( RO +

+ z1- /2 dcba /1/1/1/1 ], (12.18)

saidanac (12.15)-is msgavsad, aigeba ndobis intervali RO~

-saTvis:

[ RO~ exp(-z1-/2 dcba /1/1/1/1 ),

RO~ exp(z1-/2 dcba /1/1/1/1 )]. (12.19)

12.6. fiSeris zusti kriteriumi.

wina nawilebSi SemoTavazebuli hipoTezaTa Semowmebisa da ndob-

is intervalebis agebis meTodebi arsebiTad eyrdnoboda normalur apr-

oqsimacias. rogor moviqceT im SemTxvevebSi, roca n111

~np (1 -

11~

np ) 5

an n2 22

~np (1 -

22~

np ) 5 piroba ar sruldeba? swored amis garkvevas

eTmoba exeba winamdebare nawili. am situaciis sailustraciod ganvixi-

loT Semdegi

160

magaliTi12.4. erT-erT patara dabaSi Catarda e.w. retrospeqti-uli gamokvleva, romli mizani iyo imis garkveva moqmedebs Tu ara ad-

amianis gulsisxlZarRvTa sistemis (CVD) daavadebiT gardacvalebaze

e.w. “mlaSe” dieta. mopovebul iqna cxrilSi warmodgenili monacemebi:

gardacvalebis dietis tipi cxrili 12.9

mizezi mlaSe ara mlaSe

ara CVD 2 23 25 = n1

CVD 5 30 35 = n2

7 = m1 53 = m2 60 = n

aris Tu ara dietis tipsa da CVD –Ti gardacvalebas Soris kavSiri,

Tu isini damoukidebelia?

mocemulobidan gvaqvs, n111

~np (1-

11~

np )= 25(2/25)(1-2/25) = 46/25 =

=1.84 da n2 22

~np (1 -

22~

np ) = 35(5/35)(1-5/35) = 30/7 4.3 da rogorc

vxedavT, orive es ricxvi naklebia 5-ze. amitom wina punqtebSi gamoye-nebuli kriteriumebi ar gamogvadgeba. aseT SemTxvevebSi iyeneben fiSe-ris zust kriteriums, romelic gulisxmobs 22 cxrilSi dakvirvebu-

li mniSvnelobebis miRebis albaTobis zusti mniSvnelobis gamoTvlas,

rac Tavis mxriv, albaTobis TeoriaSi cnobilia hipergeometriuli ga-nawilebis saxeliT. mokled aRvweroT es ganawileba. davuSvaT, ori

tipis obieqtebidan, romelTa raodenobebia n1 da n2 xdeba m1 cali

obieqtis SemTxveviT amorCeva. aRvniSnoT X-iT SemTxveviTi sidide,

romelic gviCvenebs pirveli tipis obieqtebis raodenobas amorCeulm1 obieqts Soris. cxadia, rom Tavidanve unda sruldebodes piroba,

rom m1 n n1+n2, radgan n obieqtze metis amorCeva SeuZlebelia. igi-

ve SeiZleba asec gamovTqvaT, m2 n – m1 0. cxadia, rom X SemTxveviTisidide diskretulia, romelsac SeuZlia mxolod mTeli, arauaryofi-

Ti mniSvnelobebis miReba min(n1; m1)-mde. maSin am SemTxveviTi sididismier a mniSvnelobis miRebis albaToba tolia:

1

21

1

21/}{ m

nnam

nan CCCaXP

, a = 0,1,2,…, min(n1; m1). (12.20)

albaTobaTa am ganawilebas ewodeba swored, hipergeometriuli ganawi-leba.

Tu gavixsenebT 12.7 cxrilis obieqtebs da SevadarebT (12.20)

formulas, aSkarad aRmovaCenT msgavsebas da mivxvdebiT, rom albaTob-

is is zusti mniSvneloba, romelzedac zemoT iyo laparaki, swored

rom (12.20) formuliT gansazRruli albaTobaa. garda amisa, SevniSn-

oT, rom X SemTxveviTi sididis ricxviTi maxasiaTeblebia

EX = n1 m1/n da DX = n1 n2 m1 m2/(n2 (n-1)), (12.21)

161

rac kidev erTxel adasturebs, rom X SemTxveviTi sididisa maTematik-uri lodini da cxrilis (1,1) ujredis mosalodneli mniSvneloba er-

Tidaigive obieqtia.

gamoTvlebisaTvis ufro moxerxebelia albaTobis (12.20) zusti

mniSvneloba gadavweroT 22 cxrilis terminebSi Semdegi saxiT:

!!!!!

)!()!()!()!(}{

ndcba

dbcbcabaaXP

. (12.22)

axla Cven mzad varT CamovayaliboT amocanis dasma hipoTezebis

terminebSi da moviyvanoT fiSeris kriteriumi. nulovani hipoTeza ase

yalibdeba: H0 : p1 = p2, sadac p1 da p2 Sesabamisad, pirveli da meore

striqonis proporciebia. alternativa, rogorc viciT, SeiZleba iyos

ormxrivi an calmxrivi (marcxena da marjvena). amitom alternatiuli

hipoTezisaTvis gvaqvs Semdegi sami SemTxveva:

1) H1 : p1 p2; 2) H1 : p1 < p2; 3) H1 : p1 > p2. fiSeris zusti kriteriumi Zalian hgavs erTamokrefiani bin-

omuri proporciisa da puasonis intensivobisTvis martivi hipoTezis

Semowmebis zust kriteriumebs, damyarebuls p-mniSvnelobis gamoTvla-

ze. davuSvaT, cxrilis (1,1) ujredSi dgas a ricxvi. maSin:1) alternativis SemTxvevaSi p-mniSvneloba Semdegnairad iTvle-

ba:

p = 2 min(P{X a}, P{X a}, 1/2) = 2 min(P{X a},1- P{X a-1}, 1/2);2) alternativis SemTxvevaSi p-mniSvneloba Semdegnairad iTvle-

ba: p = P{X a};3) alternativis SemTxvevaSi p-mniSvneloba Semdegnairad iTvle-

ba: p = P{X a}= 1- P{X a -1}.samive SemTxvevaSi statistikuri kriteriumi ase yalibdeba:

Tu p-mniSvneloba sakmarisad mcirea (magaliTad, mcirea 0.05-ze,0.01-ze da a.S.), maSin Sedegi statistikurad mniSvnelovania da maS-

asadame, H0 : p1 = p2 hipoTeza unda uarvyoT, winaaRmdeg SemTxvevaSi,

amis safuZveli ara gvaqvs.am kriteriumis gamoyeneba cxadia, maSinacaa SesaZlebeli, roca

normaluri aproqsimacia dasaSvebia, magram erTaderTi sirTule am

kriteriumisa swored, albaTobaTa gamoangariSebaSia da amitomac amjo-

bineben normalur aproqsimacias, roca is SesaZlebelia, miT umetes,

rom aseT SemTxvevaSi orive gziT miRebuli Sedegebi sakmarisad axlo-

saa erTmaneTTan. gamoTvlebis gamartivebis mizniT, SevniSnoT, rom

P{X = k} albaTobebisaTvis, romlebic dagvWirdeba P{X a}= P{X = 0} ++ P{X = 1}+…+ P{X = a} albaTobis gamosaangariSeblad, samarTlianiaSemdegi rekurentuli Tanafardoba:

P{X = k + 1}= P{X = k}((n1-k)/(k+1)) ((m1-k)/(n2-m1+k+1)). (12.23)

162

davubrundeT 12.4 magaliTs da (12.23)-is gamoyenebiT gamovTval-oT p-mniSvneloba ormxrivi alternativisaTvis:

P{X = 1}= P{X = 0}((25-0)/(0+1)) ((7-0)/(35-7+0+1)) = 6.034 P{X = 0},saidanac P{X 1}= P{X =0}+P{X =1}=

= P{X = 0}+6.034 P{X = 0}=7.034P{X = 0}.analogiurad,

P{X = 2}=P{X =1}((25-1)/(1+1))((7-1)/(35-7+1+1))== 2.4P{X =1}=2.46.034P{X = 0}=14.483P{X = 0},

amitom

P{X 2}= P{X =0}+P{X =1}+P{X =2} == P{X = 0}+6.034P{X = 0}+14.483P{X = 0}= 21.517P{X = 0}.(12.20) formulidan gvaqvs, rom

P{X = 0}= 760

735

760

0735

025 // CCCCC

=

6059585756555453!28!

53!3534333231302928!

60!28!

53!35!

0174.030267

527

591927

1731

60595857565554

35343332313029

.

amitom

P{X 2}=21.517P{X = 0}=21.517 0.0174 0.3744;P{X 2}= 1- P{X 1}= 1-7.034P{X = 0}=1-7.0340.0174 0.878

da p-mniSvneloba tolia

p = 2min(0.3744, 0.878, 0.5) = 20.3744 = 0.7488.aqedan davaskvniT, rom p1 da p2 proporciebi mniSvnelovnad ar

gansxvavdebian da maSasadame, Cven ar SegviZlia imis Tqma, rom mlaSe

dietasa da sikvdilis mizezs Soris mniSvnelovani kavSiria.

12.7. maknemaris kriteriumi proporciebisaTvis dawyvilebul

monacemebSi.

am nawilSi Cven ganvagrZobT 22 SeuRlebis cxrilebis mimoxi-

lvas da gvainteresebs proporciebis Sedarebis sakiTxi, mxolod ara

damoukidebeli populaciebisaTvis, aramed dawyvilebuli monacemebisa-

Tvis. sxva sityvebiT rom vTqvaT, bernulis SemTxveviTi sidideebi X11,X12,…, X1n da X21, X22,…, X2n (romelTa SekrebiTacaa miRebuli Sesabami-

sad S1 = X11 + X12 + … + X1n da S2 = X21 + X22 + … + X2n dakvirvebadi bino-

muri SemTxveviTi sidideebi) ar arian wyvil-wyvilad damoukidebeli,

ris gamoc araa damoukidebeli TviTon S1 da S2 binomuri SemTxveviTi

sidideebi.

magaliTi 12.5. undodaT SeedarebinaT mkerdis kibos ori qimioT-

erapiuli reJimi masteqtomiis (mastectomy) Semdeg. amisaTvis SeirCaavadmyofTa ori jgufi da isini daawyviles iseTnairad, rom wyvilebis

163

asakebs Soris ar yofiliyo didi sxvaoba (araumetes 5 weli) da maThqonodaT TiTqmis erTnairi winaswar gansazRvruli monacemebi. amis

Semdeg, yoveli wyvilis SemTxveviT arCeul wevrs utardeboda A tipis(pirveli kviris ganmavlobaSi masteqtomiidan da ase 6 Tvis ganmavlob-

aSi), meores ki B tipis qimioTerapia (mxolod pirveli kviris ganmav-

lobaSi masteqtomiidan). avadmyofebis mkurnalobaze dakvirveba mimdina-

reobda 5 wlis ganmavlobaSi da miiRes Semdegi monacemebi:

qimioTerapiis gardaicvala 5 wlis manZilze cxrili 12.10

tipi ara ki

A 526 95 621

B 515 106 621

sul 1041 201 1242

cxadia, rom aseTnairad Sedgenili SeuRlebis cxrili A da Btipis qimioTerapiis efeqturobaze verafers gvetyvis, radgan gardac-vlilTa (an piriqiT, gadarCenilTa) raodenobebi sakmarisad axlosaa

erTmaneTTan, ufro sworad, erTmaneTTan axlosaa proporciaTa Sefase-

bebi: np1~ = 95/621=0.153 da np1

~ =106/621=0.171 da iatsis 2(1)-kriter-iumis statistikis mniSvnelobaa 0.59, rac umniSvneloa Sesabamisi po-

pulaciebis p1 da p2 proporciebis gasarCevad. garda amisa, SevniSnoT,

rom aq monacemTa saerTo raodenobaa n = 1242, rac eqsperiments ar Se-esabameba, radgan Cven vukvirdebodiT sinamdvileSi wyvilebs monacemebi-

sa, romelTa raodenobaa n = 621. amitom swori daskvnebis gasakeTeblad

saWiroa Sesamisi (swori) cxrilis ageba. mas Semdegi saxe aqvs:

gardaicvala 5

wlis manZilze

gardaicvala 5 wlis manZilze

B tipis Terapia

cxrili 12.11

A tipis Terapia ara ki

ara 510 16 526

ki 5 90 95

sul 515 106 621

am cxrilis ujredebSi Cawerili ricxvebi gviCvenebs ukve im

wyvilebis raodenobas, romlis wevrebic xuTi wlis Semdeg orive co-

cxalia ((1,1) ujredi); A tipis TerapiiT cocxalia, B-Ti ki ara

((1,2) ujredi); B tipis TerapiiT cocxalia, A-Ti ki ara ((2,1) ujre-di) da wyvilis orive wevri gardaicvala ((2,2) ujredi).

164

radgan SerCevebi araa damoukidebeli, proporciaTa Sesadarebl-

ad 2 –kriteriumi ar gamogvadgeba. aseT SemTxvevebSi iyeneben maknemar-is kriteriums. am kriteriumis aRsawerad xmaroben aseT terminebs:

wyvils, romelSic aRiricxa erTidaigive Sedegi SeTanxmebuli anu kon-kordantuli (concordant), xolo wyvils, sadac aRiricxa sxvadasvxa

Sedegi – SeuTanxmebeli, anu diskordantuli (discordant) wyvili ewod-

eba. cxadia, rom konkordantuli wyvilebi arafers gveubneba mkurnal-

obis tipis efeqturobis Sesaxeb da amitomac isini ar monawileobs

kriteriumis statistkis gamosaxulebaSi (ix. qvemoT) da piriqiT, swo-red diskordantul wyvilebSi aRmoCenili gansxvaveba metyvelebs mkur-

nalobis efeqturobaze da amitomac maknemaris statistikac maTzea age-

buli. SevniSnoT, rom diskordantuli wyvilebi ornairia, pirobiTad,

(A, BC) da (AC, B). gasagebia, rom Tu am ori tipis wyvilebis raodenob-

ebi axlosaa erTmaneTTan, maSin Zneli unda iyos imis garCeva Tu mkur-

nalobis romeli tipi romels jobia. fardobiTi sixSireebis termineb-

Si es imas niSnavs, rom TiToeuli diskordantuli tipis wyvilebis

fardobiTi sixSire axlosaa 0.5-Tan. amitomac hipoTeza da alternati-

va swored diskordantuli tipis wyvilebis albaTobis terminebSi gam-

oiTqmeba.aRvniSnoT (A, BC) tipis diskordantuli SemTxvevis moxdenis

albaToba p-Ti. maSin nulovani hipoTeza ase yalibdeba: H0 : p = 0.5, xo-lo alternativaa H1 : p 0.5. aRvniSnoT nD -Ti diskordantuli wyvil-

ebis saerTo raodenoba, xolo XD,A -Ti diskordantuli (A, BC) tipiswyvilebis raodenoba, romelic binomurad ganawilebulia parametrebiT

nD da p. amitom E XD,A = nD p da DXD,A = nD p(1-p), xolo H0 : p = 0.5hipoTezis samarTlianobis SemTxvevaSi, E XD,A = nD / 2 da DXD,A = nD / 4.aqedan gasagebia Tu rogor unda aigos kriteriumi: roca nD / 4 5, anunD 20, SegviZlia gamoviyenoT binomuri ganawilebis normaluri apro-qsimacia. winaaRmdeg SemTxvevaSi, ki p-mniSvnelobaze damyarebuli zus-

ti kriteriumi, rogorc amas vakeTebdiT binomuri ganawilebis parame-

tris Sesaxeb martivi hipoTezis Sesamowmeblad erTamokrefian amocana-

Si.

sabolood, normaluri aproqsimaciis SemTxvevaSi, anu nD 20,maknemaris (Sesworebul) kriteriumis statistikas warmoadgens sidide

TD = (|XD,A - nD / 2| - 1/2)2 /( nD / 4), (12.24)

romelsac hipoTezis samarTlianobis SemTxvevaSi, cxadia, aqvs 2(1) -ganawileba da amitom kriteriumi ase yalibdeba:

Tu TD statistikis dakvirvebuli tD = (|nA - nD / 2| - 1/2)2 /(nD / 4)mniSvnelobisaTvis, sadac nA aRniSnavs cxrilSi diskordantuli (A,BC) tipis wyvilebis raodenobas, mocemuli mniSvnelovnebis donisa-Tvis sruldeba piroba

165

tD > 21,1 , (12.25)

maSin mniSvnelovnebis doniT H0 : p = 0.5 hipoTezas uarvyofT, winaa-Rmdeg SemTxvevaSi, amis safuZveli ara gvaqvs.

Tu normaluri aproqsimaciis dauSvebelia, anu nD < 20, maSinSemdegnairad viTvliT p-mniSvnelobas

A

D

n

k

kknCp

0

5.02 , roca nA nD /2,

D

A

D

n

nk

kknCp 5.02 , roca nA > nD /2 (12.26)

da Tu is 0.05-ze (0.01-ze) naklebia, hipoTezas uarvyofT, winaaRmdeg

SemTxvevaSi, amis safuZveli ara gvaqvs.davubrundeT 12.5 magaliT. cxadia, rom nD = 21 > 20, anu norma-

luri aproqsimacia dasaSvebia. cxadia, aseve, rom nA = 5 da

tD = (|5 - 21 / 2| - 1/2)2 /(21 / 4) = 4.76. aviRoT = 0.05. maSin 295.0,1 =3.84.

radgan 4.76 > 3.84, = 0.05 mniSvnelovnebis doniT H0 : p = 0.5 hipoTe-zas uarvyofT, anu p 0.5. gasagebia isic, rom p < 0.5, anu A tipis qi-mioTerapiiT mkurnaloba jobia B-Ti mkurnalobas.

amocanebi1. populaciaSi garkveuli Tmis feris mqone mamakacebisa da qa-

lebis Tanafardoba daaxloebiT Tanabaria. satelevizio gmirebis SemT-

xvevaSi es ase ar aris. mkvlevars ainteresebda daedgina arsebobs Tu

ara kavSiri satelevizio gadacemebis personaJi qalebis Tmis fersa da

karieras Soris. cxrilSi mocemulia kvlevis Sedegebi.

Tmis feri

Q qera muqi

Pprofesiuli 36 48

kariera

K araprofesiuli 24 72

1.1 ra kriteriumi unda gamoiyenoT imis dasadgenaT arsebobs Tu arakavSiri Tmis fersa da karieras Soris?

1.2 dakavSirebulia Tu ara satelevizio personaJis samsaxuris done

misi Tmis ferTan?

2. kvlevebma aCvena, rom pirvel Svilebs, rogorc wesi, ukeTesi

Sedegebi aqvT akademiur testebSi, vidre rigiT momdevno Svilebs.

dabadebis rigi

P pirveli Svili momdevno Svili A

kreatulobis zeda mesamedi 47 29

testis qula Sua mesamedi 29 35

166

qveda mesamedi 24 36aris Tu ara kavSiri dabadebis rigsa da kreatulobas Soris mamakaceb-

Si? gamoiyeneT mniSnelovnobis 0.05 done.

3. fsiqiatriul ganyofilebaSi Semosuli axali klientebis mi-

Rebis kategorizacia moxda mTvaris fazebis mixedviT. miRebuli iyo

Semdegi monacemebi:

savse mTvare axali mTvare pirveli meoTxedi mesame meoTxedi

62 50 60 56

gansxvavebulia Tu ara miRebuli klientebis raodenoba mTvaris fazis

mixedviT? gamoiyeneT Tanxmobis kriteriumi.

4. ra SemTxvevebSi ixmareba araparametruli kriteriumebi?5. ra saxis monacemebis analizi aris xi-kvadrat kriteriumisaT-

vis Sesaferisi?

6. ra aris warmodgenili niSanTa SeuRlebis cxrilis svetebSi?

7. ra aris warmodgenili niSanTa SeuRlebis cxrilis striqone-

bSi?

8. mkvlevari akvirdeboda im avtomobilebis raodenobas, romleb-

ic “sdeq” niSanze ar Cerdebodnen.

K kviris dRe

samSabaTi oTxSabaTi xuTSabaTi

D dilis 10:00-dan 38 46 58 dro

SuadRis 2:00-dan 46 54 63

3:00-mde

aris Tu ara aseTi qcevis sixSire damokidebuli dRis drosa da kvir-

is dReze?

9. aris mosazreba, rom mamakacebis mier qalebis mosaZebnad gan-

Tavsebul gancxadebebSi ufro xSirad aRniSnulia wonis moTxovna, vi-

dre qalebis mier mamakacebis mosaZebnad ganTavsebul gancxadebebSi. am

hipoTezis Sesamowmeblad gamoikvlies piradi gancxadebebis garkveuli

raodenoba da moaxdines maTi kategorizacia gancxadebis avtoris sqes-isa da gancxadebaSi sasurveli wonis aRniSvna-araRniSvnis mixedviT.

aRniSnuli iyo Tu ara sasurveli wona

diax ara

mamakacis gancxadeba 89 186

qalis gancxadeba 6 219

arsebobs Tu ara am monacemebs Soris is kavSiri, rasac mosazreba var-

audobs?

10. fsiqologma SemTxveviT SearCia 8 vaJi da maT gaacno ganaT-

lebis programa. 6 kviris Semdeg man am vaJebs Seavsebina kiTxvari al-koholis moxmarebis Sesaxeb. Aamavdroulad man SearCia 10 sxva vaJi,

167

romlebic ar gascnobian alkoholis moxmarebis winaaRmdeg programasda maTac Seavsebina kiTxvari. yoveldRiurad moxmarebuli alkoholis

raodenoba unciebSi Semdegnairi aRmoCnda:

gaiara programa – 0.31 0.53 0.58 0.14 0.16 0.52 0.53 0.02

ar gauvlia programa – 0.41 0.63 1.14 0.21 0.89 0.55 0.89 0.91 0.08 0.59

man-uitnis kriteriumis gamoyenebiT SeamowmeT moaxdina Tu ara alkoh-

olis moxmarebis sawinaaRmdego saganmanaTleblo programam gavlena im

adamianebis mier alkoholis moxmarebaze, vinc am programas gaecno.

l e q c i a 13.

12.8. SerCevis moculobis gansazRvra da kriteriumis simZlavre

ori binomuri proporciebis Sedarebisas.

am nawilSi Cven SevexebiT imave tipis amocanas, rogoric ganxi-

luli iyo 9.5 punqtSi: SerCevis im minimaluri moculobis gansazRr-

as, romelic saWiroa mocemuli mniSvnelovnebis donisas dasaxelebuli

simZlavris misaRwevad ori normaluri populaciis saSualoTa tolo-

bis Sesaxeb hipoTezaTa garCevis amocanaSi. gansxvaveba cxadia, isaa,

rom axla ukve Sesadarebeli populaciebi binomuria da ara normalu-

ri.

12.8.1. damoukidebeli populaciebis SemTxveva.

ganvixiloT jer damoukidebeli populaciebis SemTxveva. rog-

orc gvaxsovs, hipoTeza da alternativa, Sesabamisad, ase yalibdeba:

H0 : p1= p2 da H1 : p1 p2. garda amisa, rogorc 12.2 punqtidan gvaxs-

ovs,11

~np ~N(p1; p1(1- p1)/n1) da

22~

np ~ N(p2; p2(1-p2)/n2) da11

~np -

22~

np

sxvaoba warmoadgens damoukidebeli normaluri SemTxveviTi sidideebis

sxvaobas, romelic isev normalurad ganawilebuli alternativis sama-

rTlianobis SemTxvevaSi. misi saSualo tolia p1 - p2 sididis. amitom

kriteriumis simZlavris gamoTvla mniSvnelovnebis donis dros xde-ba Semdegi formulis saSualebiT:

)/)~1(~/)~1(~)/1/1()~1(~|~~|

1222111

21,,2/121

2211

212121

nppnpp

nnppzpp

nnnn

nnnnnn , (12.27)

sadac11

~np x1 / n1,

22~

np x2 / n2 da21

2211

21

21,

21

21

~~~

nn

pnpn

nn

xxp

nn

nn

.

168

SerCevaTa moculobebis dadgenisas ki, gvinda, rom simZlavre ar

iyos (1-)-ze naklebi. garda amisa, vTqvaT, k n2 / n1. maSin (12.27)-dan

saWiro SerCevis moculobisaTvis samarTliania Semdegi formula:

1 1 2 2

1 2

21 / 2 1 1 2 2 1

1 21 2

( (1 )(1 1/ ) (1 ) (1 ) / ) )

( )k k n n n n

n n

p p k z p p p p k zn

p p

(sadack

pkpp

nn

k

1

~~~ 21 21

, n2 = k n1). (12.28)

magaliTi 12.6. davuSvaT daigegma gamokvleva samkurnalo da qir-

urgiuli Carevebis Sesadareblad bavSvebSi, romlebsac pirveli sami

wlis ganmavlobaSi aqvT otitis (yuris anTebis, OTM-otitis media) met-ismetad xSiri SemTxvevebi. mkurnalobaTa warmatebis SemTxvevebi Sesab-

amisad, 50% da 70%-s Seadgens. warmatebaSi igulisxmeba OTM-is ara-

umetes erTi SemTxvevisa samedicino Carevidan erTi wlis ganmavloba-

Si. realisturia TiToeul jgufSi 100-100 pacientis CarTva. rogori

iqneba kriteriumis simZlavre = 0.05 mniSvnelovnebis donisaTvis?

amoxsna. mocemulobis Tanaxmad,11

~np = 0.5,

22~

np = 0.7 da radgan

n1=n2=100, amitom gvaqvs:

21 ,~

nnp =(n111

~np + n2

22~

np )/(n1+n2)=(11

~np +

22~

np )/2=(0.5+0.7)/2= 0.6.

vinaidan z1- 0.05/2 = z0.975 =1.96, (12.27) formulis Tanaxmad, gveqneba:

83.0)947.0(100/)7.01(7.0100/)5.01(5.0

)100/1100/1)(6.01(6.096.1|7.05.0|1

.

maSasadame, 100-100 pacientis CarTvisas, ori tipis mkurnalob-

ebs Soris mniSvnelovani ( = 0.05) gansxvavebis daWera SesaZlebelia

83% SemTxvevaSi.magaliTi 12.7. davuSvaT pirobebi iseTivea, rogorc wina magali-

TSi. iq Cven realisturad CavTvaleT 100-100 pacientis CarTva gamokv-levaSi, magram zogierTi mizezis gamo eqsperimentis gegmis mixedviT

warmarTva ar xerxdeba. aRvniSnoT, Tundac is faqti, rom bavSvebis

mSobelTa 75% uars ambobs qirurgiul Carevaze da amis gamo, Cven Se-

gviZelia mxolod n2 = n1/4 monacemis mopoveba. ramxela unda aviRoT Se-

rCevaTa moculobebi, rom = 0.05 mniSvnelovnebis donisaTvis kriter-iumis simZlavre iyos aranakleb 83%-isa?

amoxsna. mocemulobiT isev gvaqvs11

~np = 0.5,

22~

np = 0.7 magram ra-

dgan n1 n2 da n2 = n1/4 e.i. k =1/4 da amitom

kp~ = (11

~np + k

22~

np )/(1+k) = (0.5 + 0.250.7)/(1+0.25) = 0.54.

normaluri ganawilebis kvantilebis cxrilebidan vpoulobT, rom z0.975

=1.96, z0.83 = 0.95 da (12.28) formulis mixedviT, sabolood gvaqvs:

169

174.251

)7.05.0(

95.043.07.05.05.096.1)25.0/11(46.054.02

2

1

n ,

saidanac davaskvniT, rom n1 = 252 da n2 = 252/4 = 63.

12.8.2. dawyvilebuli monacemebis SemTxveva.

ganvixiloT axla dawyvilebuli monacemebis SemTxveva, romlis-

aTvisac populaciebi aRaraa damoukidebeli. rogorc 12.5 punqtidanviciT, am SemTxvevaSi gamoiyeneba maknemaris kriteriumi, romelic Tav-

is mxriv, kerZo SemTxveva iyo erTamokrefiani amocanisa binomuri para-

metris Sesaxeb. amitom gavixsenoT (9.12) da (9.13) formulebs me-8

leqciidan, romlebsac kriteriumis simZlavrisa da SerCevis moculob-

isaTvis, Sesabamisad, Semdegi saxe hqondaT

npp

ppz

pp

pp

)1(

||

)1(

)1(1

00

102/

11

00 , (9.12’)

2

00

1112/12

10

00

)1(

)1(

)(

)1(

pp

ppzz

pp

ppn . (9.13’)

maknemaris kriteriumis SemTxvevaSi hipoteturi p = p0 =1/2, xo-lo alternativis rolSi aviRoT p1 = pA – A tipis diskordantuli

wyvilebis wili saerTo diskordantul wyvilebs Soris. maSin kriter-iumis simZlavrisa da diskordantuli monacemebis raodenobisaTvis, Se-

sabamisad, gveqneba:

)1(2

|12|1 2/

AA

DA

pp

npz , (12.29)

2

2

12/1

)2/1(4

)1(2

A

AAD p

ppzzn

. (12.30)

imisaTvis, rom gamovTvaloT ara diskordantuli wyvilebis, ar-

amed wyvilebis saerTo raodenoba, aRvniSnoT pD-Ti albaToba imisa,

rom SemTxveviT amorCeuli wyvili iqneba diskordantuli. maSin cxa-

dia, rom pD = nD / n, saidanac nD = n pD da (12.30) formulaSi CavsvaTes mniSvneloba. miviRebT:

2

2

12/1

)2/1(4

)1(2

AD

AA

pp

ppzzn

. (12.31)

aseve gasagebia, rom Tu Cven gvainteresebs ara wyvilebis, aramed

individebis raodenoba, maSin ukanaskneli formuliT gamoTvlili n si-

dide unda gavamravloT 2-ze.magaliTi 12.8. davuSvaT pirobebi iseTivea, rogorc 12.5 magali-

TSi A da B tipis qimioTerapiiT mkurnalobis efeqtis Sedarebaze da

170

aRmoCnda, rom 400 dawyvilebuli monacemidan, wyvilebis 85% konkor-dantulia, anu maTi wevrebi “iqcevian” erTnairad: orive gadarCa, an

orive gardaicvala. garda amisa, (A,BC) tipis diskordantuli wyvile-

bis raodenoba Seadgens diskordantuli wyvilebis saerTo raodenobis

2/3-s. rogoria maknemaris kriteriumis simZlavre = 0.05 mniSvnelo-

vnebis donisaTvis?

amoxsna. mocemulobis Tanaxmad, = 0.05, pD = 1-0.85 = 0.15, pA ==2/3 da n = 400. radgan z0.025 = -z0.975 = -1.96, amitom (12.29) formuliT,

gveqneba:

745.0)66.0()3/1()3/2(2

15.0400|13/22|96.11

.

maSasadame, 400 dawyvilebuli monacemisaTvis, statistikuri mniSvnel-

ovnebis aRmoCena SeiZleba eqsperimentebis daaxloebiT 75%-Si.

magaliTi 12.9. davuSvaT igivea rac wina magaliTSi, magram Cven

gvainteresebs ufro maRali simZlavris, 90%-is, miRweva. ramden adam-

ianze dakvirvebaa eqsperimentSi saWiro?

amoxsna. mocemulobis mixedviT, = 0.05, pD = 1-0.85 = 0.15, pA ==2/3 da 1- = 0.9. radgan z0.975 = 1.96 da z0.9 = 1.28, amitom (12.31) form-uliT,

32.601

0167.0

1668.3

)2/13/2(15.04

3/13/228.1296.1)(

2

2

2

wyvilebisn .

maSasadame, sabolood, n(adamianebis) = 2602 = 1204.

12.9. SeuRlebis r c cxrilebi.

wina punqtebSi yvelgan laparaki iyo SeuRlebis 22 cxrileb-

ze, anu yoveli Sesaswavl sidides mxolod ori kategoria gaaCnda.

xSirad erT an orive Sesaswavl sidides SeiZleba orze meti katego-

ria gaaCndes da amitomac Cndeba SeuRlebis 22 cxrilebis ganzogad-

ebis saWiroeba SeuRlebis r c cxrilebamde, anu cxrilebamde, romel-

Sic Sesaswavli sididdeebis kategoriebi warmodgenilia r striqonsada c svetSi.

magaliTi 12.10. gavixsenoT 12.1 magaliTi, sadac laparaki iyo

mkerdis kibosa da qalis pirveli mSobiarobis asakze. im magaliTSierT-erTi dakvirvebadi sidide – qalis asaki sul or kategoriad iyo

dayofili: pirveli mSobiaroba hqondaT 30-ze meti an toli wlis asa-

kSi da pirveli mSobiaroba hqondaT 30 welze qvemoT. davuSvaT axla,

rom Cven ufro dawvrilebiT viciT Sesaswavli qalebis pirveli mSobi-

arobis asakebi, kerZod, SesaZlo asakobrivi intervali dayofilia 5kategoriad da CvenTvis cnobilia TiToeul kategoriaSi qalebis rao-

171

denobebi orive jgufis qalebisaTvis, visac aqvs mkerdis kibo da visaces daavadeba ara aqvs. Sedegebi warmodgenilia Semdeg cxrilSi:

kategoria qalisAasaki pirveli mSobiarobisas cxrili 12.12

kibo < 20 20-24 25-29 30-34 35 sul

aqvs 320 1206 1011 463 220 3220Ara aqvs 1422 4432 2893 1092 406 10245sul 1742 5638 3904 1555 626 13465

kibos %-ebi 18.4 21.4 25.9 29.8 35.1 23.9Cven gvinda davadginoT damokidebuleba pirveli mSobiarobis asaksa da

mkerdis kibos Soris. rogor unda gakeTdes es?

rogorc vxedavT, saqme gvaqvs SeuRlebis 25 cxrilTan da bun-

ebrivia, am cxrilsac davuZaxoT dakvirvebuli SeuRlebis cxrili. 22cxrilis SemTxvevis msgavsad, aqac unda avagoT Sesabamisi mosalodne-li SeuRlebis cxrili da bolos es cxrilebi SevadaroT. imisaTvis

rom gamovTvaloT ukansknelis (i,j) ujredSi Casaweri Eij elementi, uf-

ro swored, ki misi Sefaseba ijE~, i =1,2, j=1,2,3,4,5 (ix. qvemoT zogadi

SemTxveva), viqceviT isev ise, rogorc 22 cxrilebisaTvis: i-uristriqonisa da j-uri svetis marginaluri mniSvnelobebis namravls

vyofT cxrilis elementebis saerTo raodenobaze. magaliTad,

11

~E = 3220 1742/13465 416.6, 12

~E = 3220 5638/13465 1348.3,

21

~E = 10245 1742/13465 1325.4, 25

~E = 10245 626/13465 476.3.

yvela ijE~ elementis gamoTvlis Semdeg miviRebT, rom mosalodn-

el SeuRlebis cxrils aqvs Semdegi saxe:

kategoria qalisAasaki pirveli mSobiarobisas cxrili 12.13

kibo < 20 20-24 25-29 30-34 35 sul

aqvs 416.6 1348.3 933.6 371.9 149.7 3220Ara aqvs 1325.4 4289.7 2970.4 1183.1 476.3 10245sul 1742 5638 3904 1555 626 13465sazogadod, unda iTqvas, rom SeuRlebis r c cxrilebic 22

cxrilebis msgavsad, gamoiyeneba ori ZiriTadi tipis amocanebSi: erTis

mxriv, erTi SerCeva raime populaciidan dayofilia r da c kategorie-bad ori sxvadasxva niSnis mixedviT da ainteresebT am niSnebis damou-

kideblobis sakiTxi (niSanTa damoukideblobis amocana), xolo meore

tipis amocanebSi, mocemulia dakvirvebebi c sxvadasxva populaciaze,

romlebic erTidaimave, garkveuli niSnis mixedviT klasificirebulia

or alternatiul (atributul) kategoriad da ainteresebT, aris Tu

ara sxvadasxva populaciis monacemTa proporciebi erTnairi (erTgvar-

ovani) yoveli konkretuli kategoriisaTvis (populaciaTa erTgvarov-nebis amocana).

172

dakvirvebuli SeuRlebis cxrilebs (ricxvebis doneze) oriveSemTxvevaSi msgavsi forma aqvT, magram isini, gaazrebuli SemTxveviTi

sidideebis TvalsazrisiT, sxvadasxva Sinaarsis matarebelia. kerZod,

arsebiTia is Tu, ramdeni “bma” aqvs cxrilis ujredebSi moTavsebul

SemTxveviT sidideebs.

niSanTa damoukideblobis amocanaSi dakvirvebuli SeuRlebis

cxrils Semdegi saxe aqvs:

kategoriebi II niSniT cxrili 12.14kategoriebi IniSniT 1 2 . . . c

sul

1 O11 O12 . . . O1c 1O

2 O21 O22 . . . O2c 2O

. . .

r Or1 Or2 . . . Orc rO

sul 1O 2O . . . cO O

sadac

c

jiji OO

1

, i = 1,2,…,r;

r

iijj OO

1

, j = 1,2,…,c da

O

r

i

c

jijO

1 1

.

SevniSnoT, rom O = n, sadac n-iT aRniSnulia SerCevis mocul-

oba. ase, rom Oij SemTxveviT sidideebs gaaCniaT erTaderTi bma:

r

i

c

jijO

1 1

= n. (12.32)

vnaxoT axla rogori saxe aqvs imave cxrils populaciaTa erT-

gvarovnebis amocanaSi:

SerCeva cxrili 12.15kategoria

1 2 . . . csul

1 O11 O12 . . . O1c 1O

2 O21 O22 . . . O2c 2O


sadac kvlav

c

jiji OO

1

, i = 1,2; jjj OOO 21 , j = 1,2,…,c da

O

r

i

c

jijO

1 1

, magram aq ukve jO = nj, j = 1,2,…,c, sadac nj aRniSnavs j-

uri SerCevis moculobas da maSasadame, Oij SemTxveviT sidideebs gaaCn-

iaT ukve c bma:O1j + O2j, j = 1,2,…,c. (12.33)

(12.32) da (12.33) formulebiT gamoTqmuli kavSirebi Oij SemTxv-eviT sidideebs Soris, sxvanairad SeiZleba asec gamovTqvaT: pirvel

173

SemTxvevaSi, marginaluri striqonis mxolod bolo svetSi Cawerili

ricxvia ( O ) fiqsirebuli, maSin, roca meore SemTxvevaSi, mTeli mar-

ginaluri striqonia fiqsirebuli.

aRvniSnoT pij-Ti, i = 1,2,…,r, j = 1,2,…,c, (12.14) cxrilis (i, j)-urujredSi moxvedris albaToba. pirveli tipis amocanebSi es iqneba alb-

aToba imisa, rom populaciis SemTxveviT amorCeuli elementi moxvdeba

I niSnis i-ur da II niSnis j-ur kategoriebSi. garda amisa, aRvniSnoTmarginaluri albaTobebi ip da jp -Ti. maSasadame, ip aRniSnavs pop-

ulaciis SemTxveviT amorCeuli elemntis I niSnis i-ur kategoriaSimoxvedris albaTobas, imis miuxedavad II niSnis romel kategoriaSi

moxvda is da aseve, jp populaciis SemTxveviT amorCeuli elementis

II niSnis j–ur kategoriaSi moxvedris albaTobaa, imis miuxedavad IIniSnis romel kategoriaSi moxvda is.

c

jiji pp

1

, i = 1,2,…,r da

r

iijj pp

1

, j = 1,2,…,c. (12.34)

vnaxoT axla, Tu rogor yalibdeba nulovani hipoTeza da alte-

rnativa niSanTa damoukideblobis SemTxvevaSi da rogori saxe aqvs mo-salodneli SeuRlebis cxrils. rogorc albaTobis Teoriidan gvaxs-

ovs, ori xdomilobis damoukidebloba niSnavda maTi erTdroulad mox-

denis albaTobis tolobas calkeuli xdomilobis albaTobaTa namrav-

lTan. amitom yoveli (i,j) ujredisaTvis (sul cota) unda sruldebo-

des piroba

pij = ji pp , i = 1,2,…,r, j = 1,2,…,c. (12.35)

rogorc viciT, Eij = EOij = npij, xolo damoukideblobis hipoTe-

zis samarTlianobisas, Eij = npij = O ji pp . magram roca Cven ar vic-

iT ip da jp albaTobebi, maT magivrad viRebT Sesabamis Sefasebebs

ip~ = iO / O da jp~ = jO / O da mosalodneli sidideebis rolSi gamo-

dis, is razec laparaki iyo 12.10 magaliTSi mosalodneli SeuRlebis

(12.13) cxrilis agebis dros, anu ijE~

= iO jO / O . amitom mosalodne-

li SeuRlebis cxrils aqvs Semdegi saxe:

kategoria II niSniT cxrili 12.15kategoria

I niSniT 1 2 . . . csul

1 11

~E 12

~E . . . cE1

~1O

2 21

~E 22

~E . . . cE2

~2O

. . .

r 1

~rE 2

~rE . . . rcE

~rO

174


vnaxoT axla, Tu rogor yalibdeba nulovani hipoTeza da alte-

rnativa populaciaTa erTgvarovnebis SemTxvevaSi. rogorc zemoT aRvn-

iSneT, am SemTxvevaSi gvainteresebs, aris Tu ara sxvadasxva populaci-

is monacemTa proporciebi erTnairi orive kategoriisaTvis. cxadia, sa-kmarisia SevamowmoT es hipoTeza mxolod erT-erTi kategoriisaTvis,

radgan meoreSi is avtomaturad igive pasuxs mogvcems. ase, rom nulo-

vani hipoTezaa H0 : p11 = p12 = … = p1c. alternativaa, rom ori albaTo-

ba mainc gansxvavebulia.

statistikuri kriteriumebis asagebad vmsjelobT isev ise, rog-

orc 22 cxrilis dros: ramdenadac ufro daSorebuli iqneba SeuRl-

ebis dakvirvebuli da mosalodneli cxrilis ujredebSi moTavsebuliSesabamisi ricxvebi, hipoTezis samarTlianobis miT ufro naklebi San-

sia. am ricxvebis siaxlovis sazomad ki (i,j) ujredisaTvis kvlav viyen-

ebT (Oij - Eij)2/ Eij sidides, i = 1,2,…,r, j = 1,2,…,c, xolo kriteriumis

statistikad viRebT maT jams, anu

r

i

c

jijijijrc EEOT

1 1

2 /)( , (12.36)

romelsac mtikcdeba, rom H0 hipoTezis samarTlianobis dros aqvs

2((r-1)(c-1))-ganawileba. garda amisa, erTgvarovnebis amocanaSi yvelaf-eri igivea, ubralod, am SemTxvevaSi, r = 2.

sabolood, statistikuri kriteriumi aseTnairad yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis Trc statistik-

is dakvirvebuli trc mniSvneloba metia 2(r-1)(c-1),1- sidideze, maSin H0

hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi, amis safuZveli ara

gvaqvs.

Sesabamisi p-mniSvneloba tolia p = P{2(r-1)(c-1) > trc}.

erTaderTi, rac am kriteriumis gamoyenebisas unda gvaxsovdes

isaa, rom daculi iyos Semdegi pirobebi: a) mosalodneli SeuRlebis

cxrilis arcerT ujredSi ar unda iyos 1-anze naklebi ricxvi da b)5-anze naklebi sididis ujredebis raodenoba ar unda aRematebodes

mosalodneli SeuRlebis cxrilis ujredebis saerTo raodenobis mex-

uTed nawils, anu r c/5-s.davubrundeT 12.10 magaliTs da vnaxoT, ras mogvcems kriteriu-

mi. SevniSnoT, rom radgan Cvens SemTxvevaSi, r = 2 da c = 5, Cveni amoca-na 25 cxrilis tipis amocanaa da amitom nulovan hipoTezaze SegviZ-lia vilaparakoT niSanTa damoukideblobis (mkerdis kibo da pirveli

mSobiarobis asaki) terminebSic da 5 damoukidebeli binomuri popula-

ciis erTgvarovnebis (sxvadasxva asakobrivi intervalis qalebisaTvis

mkerdis kibos proporciebis) terminebSic. 12.13 cxrilidan vxedavT,

175

rom yvela mosalodneli sidide 5-ze. amitom 12.12 da 12.13 cxrile-bis saSualebiT gamovTvaloT T2,5 statistikis dakvirvebuli t2,5 mniSv-

neloba:t2,5 = (320-416.6)2/416.6+(1206-1348.3)2/1348.3+(1011-933.6)2/933.6+

+(463-371.9)2/371.9 +(220-149.7)2/149.7 = 130.3.2-ganawilebis cxrilebidan vpoulobT, rom

2(2 -1)(5 -1),1-0.001 = 2

4, 0.999 = 18.47.radgan 2

4, 0.999 = 18.47 < 130.3 = t25, = 0.001 mniSvnelovnebis doniT ua-

rvyofT H0 hipoTezas orive midgomisaTvis da davaskvniT, rom mkerdis

kibo da pirveli mSobiarobis asaki mniSvnelovnad dakavSirebuli sidi-

deebia (damoukideblobis SemTxvevaSi) da mkerdis kibos proporciebi

ar aris erTnairi yvela asakobrivi intervalisaTvis (erTgvarovnebisSemTxvevaSi).

ukanasknel SemTxvevaSi, sainteresoa kidev erTi sakiTxi: mainc

romeli proporciebia mniSvnelovnad gansxvavebuli da romeli ara? an

rogoria proporciebis cvalebadobis tendencia (trendi) erTi asakob-rivi intervalidan meoreze gadasvlis dros? 12.12 cxrilis bolo

striqonSi miwerili iyo kibos SemTxvevebis procentuli maCveneblebi,

romelic asakobrivi intervalis zrdasTan erTad izrdeba. sworia Tu

ara davaskvnaT, rom Sesabamisi populaciebis parametrebic aseve iqcevi-

an, Tu es SemTxveviT moxda? populaciebis parametrebis trendis aseTi

yofaqcevebis aRmosaCenad SemoaqvT e.w. qulis maCvenebeli Si, romelicwarmoadgens i-uri jgufis raime kerZo ricxviT atributs, magaliTad,

jgufis nomers, an jgufis saSualo asaks da a.S. SevadginoT aseTi si-

dideebi:

c

jjjj SSOOOOA

111 )()/( (sadac

OSOS

c

jjj /

11 ), (12.37)

OSOSOOOOOB

c

jjj

c

jjj /)/1(/

2

11

1

2111 . (12.38)

mtkicdeba, rom mzardi (an klebadi) trendis ar arsebobis SemT-

xvevaSi T=A2/B statistikas aqvs 2(1)-ganawileba da amitom statisti-

kuri kriteriumi ase yalibdeba:

Tu mocemuli mniSvnelovnebis donisaTvis T = A2/B statistik-

is dakvirvebuli t mniSvneloba metia 21,1- sidideze, maSin mniSvne-

lovnebis doniT vaskvniT, rom trendi arsebobs, winaaRmdeg SemTxveva-

Si, amis safuZveli ara gvaqvs. garda amisa, Tu A > 0, maSin proporcie-bi mzardia (mzardi S qulebiT) da Tu A < 0, maSin proporciebi kleba-

dia (mzardi S qulebiT). Sesabamisi p-mniSvneloba tolia p = P{21 > t}.

aqac gvWirdeba imis damateba, rom es kriteriumi gamodgeba maSin, roca

5)/( 2121 OOOO .

176

vnaxoT ras mogvcems es kriteriumi Cveni magaliTis SemTxvevaSi.gamovTvaloT A da B gamosaxulebebis ricxviTi a da b mniSvnelobebi

(12.37) da (12.38) formulebis mixedviT, Si = 1,2,3,4,5 qulebis dros:

a=3201+12062+10113+4634+2205–-3220[17421+56382+39043+15554+6265]/13465=567.16,

b=0.2390.761{174212+563822+390432+155542+62652--[17421+56382+39043+15554+6265]2/13465}= 2493.33

dabolos,

t = a2/b = 567.162/2493.33 = 129.01.2-ganawilebis cxrilebidan vpoulobT, rom

2 1,1-0.001 = 2

1, 0.999 = 10.83.radgan 2

1, 0.999 = 10.83 < 129.01 = t, = 0.001 mniSvnelovnebis doniT ua-

rvyofT H0 hipoTezas da davaskvniT, rom proporciebSi trendi arseb-

obs da radgan a > 0, sabolood vaskvniT, rom pirveli mSobiarobis as-

akis zrdasTan erTad mkerdis kibos proporcia izrdeba.

12.10. Tanxmobis 2 kriteriumi.

am punqtSi Cven SevexebiT 2 kriteriumis gamoyenebas kidev erTmniSvnelovan amocanaSi, romelsac Tanxmobis amocanas uwodeben. imisaT-vis, rom gavigoT Tu ra Tanxmobazea laparaki gavixsenoT, rom wina

leqciebSi, exeboda isini parametrebis Sefasebis, Tu hipoTezebis Semo-

wmebis amocanebs, Cven yvelgan vuSvebdiT, rom SerCevis Sesabamisi pop-

ulaciis ganawilebis saxe cnobilia. axla Cven gvainteresebs, ramdenad

sworia daSveba vTqvaT, populaciis normalurobis Sesaxeb? eTanxmeba

SerCeva am daSvebas Tu ara? swored am kiTxvebze pasuxis gacemas emsa-

xureba es punqti. sakiTxi rom marTlac mniSvnelovania, cxadia, radganTu Tavidanve gavakeTeT araswori daSveba modelis Sesaxeb, maSin raime

amocanis amoxsnaze laparakic zedmetia mcdari modelis CarCoebSi.

magaliTi 12.11. cxrilSi mocemulia 30-69 wlis 14736 adamianisdiastoluri sisxlis wnevebis rva jgufad dayofili monacemebi, rome-

lTa SerCeviTi saSualo da dispersia tolia nx 80.68, 2ns 144. Cveni

amocanaa SevamowmoT, Tu ramdenad eTanxmeba SerCeviTi mniSvnelobebi

populaciis normalurobas = nx da 2 = 2ns parametrebiT. rogor unda

gakeTdes es?

jgufi dakvirvebuli sixSire mosalodneli sixSire

< 50 57 78 50, < 60 330 547 60, < 70 2132 2127 70, < 80 4584 4283 80, < 90 4604 4479

177

90, < 100 2119 2431 100, < 110 659 684

110 251 107sul 14736 14736

cxrilis bolo svetSi mocemulia jgufebis sixSireebi daSveb-

uli modelis mixedviT. es imas niSnavs, rom magaliTad, ricxvi 547 bo-

lo svetis meore striqonSi miRebulia aseTnairad: 547 = n p2 = 14736 p2, sadac pi aris modelis Sesabamisi N(80.68, 144) SemTxveviTi sidid-is i -ur jgufSi moxvedris albaToba, anu p2-saTvis

p2 = P{50 N(80.68; 144) < 60}= ((60-80.68)/12) - ((50-80.68)/12) == (-1.72) - (-2.56) 0.0424 – 0.0053 = 0.0371.

marTlac, Tu gavamravlebT SerCevis moculobas ukanasknel ric-

xvze, miviRebT, 14736 0.0371 547. analogiurad iangariSeba bolosvetSi mdgomi danarCeni ricxvebic.

2 statistika aigeba wina punqtebis msgavsad. gadaxris sazomad

yvela jgufisaTvis ganvixiloT (O - E)2/E sidide da avjamoT isini.mtkicdeba, rom miRebul sidides, rogorc SemTxveviT sidides, aqvs

2(g - k - 1)-ganawileba, sadac g aRniSnavs jgufebis raodenobas, xolok -- mosalodneli mniSvnelobebis gamosaTvlelad gamoyenebuli Sefase-

buli parametrebis raodenobas. kriteriumi ki sabolood ase yalibd-

eba: Tu dasaxelebuli mniSvnelovnebis donisaTvis

g

iiii EEOT

1

2 /)( (12.39)

statistikis dakvirvebuli t mniSvneloba metia 2g-k-1,1- sidideze, maS-

in mniSvnelovnebis doniT hipoTezas uarvyofT, winaaRmdeg SemTxvev-

aSi, amis safuZveli ara gvaqvs. cxadia, Sesabamisi p-mniSvneloba iTvl-

eba formuliT p =P{2g - k - 1> t}. SezRudva aqac igivea, rom a) mosalo-

dneli sixSireebidan arcerTi ar unda iyos 1-anze naklebi da b) 5-an-ze naklebi sididis mosalodnel sixSireTa raodenoba ar unda aRemat-

ebodes maTi saerTo raodenobis mexuTeds, anu g/5-s.12.11 magaliTSi cxadia, g = 8 da radgan Cven SevafaseT normal-

uri modelis maTematikuri lodinica da dispersiac, amitom k = 2 damaSasadame, Tavisuflebis xarisxi T statistikisa tolia g - k –1= 8-2-1 = 5. 2(5)-ganawilebis cxrilebidan = 0.001-saTvis vpoulobT, rom

2g - k - 1, 1- = 2

5, 0.999 = 20.52. gamovTvaloT T statistikis dakvirvebuli

ricxviTi mniSvneloba:t = (57-78)2/78+(330-547)2/547+(2132-2127)2/2127+(4584-283)2/4283 + (4604--4479)2/4479+(2119-2431)2/2431+(659-684)2/684+(251-107)2/107 = 350.2.

178

miRebuli mniSvnelobis Sedareba cxrilis dasaSveb mniSvnelobasTan

gvaZlevs: 25, 0.999 = 20.52 < 350.2 = t da maSasadame, = 0.001-mniSvnelovn-

ebis doniT, SerCevis modelTan Tanxmobis hipoTezas uarvyofT.

SevniSnoT, rom es meTodi gamodgeba ara mxolod modelis norm-

alurobis Sesamowmeblad, aramed nebismieri sxva modelebisaTvisac. er-

TaderTi gansxvaveba i-ur jgufSi moxvedris, modelis Sesabamisi pi

albaTobebis gamoTvlis wesSi iqneba.

amocanebi1. 1985 wels ikvlevdnen kavSirs IUD kontraceptivis xmarebasa

da uSvilobas Soris da aRmoCnda, rom 283 uSvilodan 89 garkveuli

drois ganmavlobaSi xmarobda am kontraceptivs, xolo sakontrolo

jgufis 3833 qalidan mas xmarobda 640 qali.

1.1. normaluri aproqsimaciis saSualebiT gaarkvieT aris Tu ara mniSv-nelovani gansxvaveba kontraceptivis momxmarebelTa or jgufs Soris;

1.2. aageT SeuRlebis dakvirvebuli cxrili;

1.3. gaarkvieT 1.1-Si dasmuli sakiTxi SeuRlebis cxrilis gamoyenebiT;

1.4. SeadareT 1.1-sa da 1.3-Si miRebuli Sedegebi;

1.5. aageT 95%-iani ndobis intervali kontraceptivis momxmareblebisa

da aramomxmareblebis proporciaTa sxvaobisaTvis sakontrolo da Ses-

aswavl jgufebSi;

1.6. gamoTvaleT Sansebis fardoba kontraceptivis momxmareblebs Sor-

is uSvilo da sakontrolo jgufebSi;

1.7. aageT 95%-iani ndobis intervali Sansebis fardobis WeSmaritimniSvnelobisaTvis 1.6 amocanaSi.

2. davuSvaT 1 amocanaSi ufro dawvrilebiTaa cnobili kontrac-

eptivis moxmarebis xangrZlivobebi TveebSi da es monacemebi warmodgen-

ilia Semdegi cxrilis saxiT:

IUD-is moxmarebis xangrZlivobebi TveebSi

jgufi < 3 3, < 18 18, < 36 36Sesaswavli 10 23 20 36sakontrolo 53 200 168 219

2.1. aris Tu ara oTxive jgufis proporciebi erTnairi?

2.2. davuSvaT jgufebs qulebad mivawereT ricxvebi 1, 2, 3, 4. izrdeba

Tu klebulobs proporciebi IUD-is moxmarebis xangrZlivobis zrdasT-

an erTad?

3. undodaT SeedarebinaT A da B tipis wamlebis efeqturoba

Tormetgoja nawlavis wylulze. amisaTvis pacientebi daawyviles asak-

is, sqesisa da klinikuri monacemebis mixedviT. 200 dawyvilebuli mon-

acemidan aRmoCnda, rom 89-ze orive wamalma dadebiTad imoqmeda, 90-ze

179

arcerTma wamalma ar iqonia gavlena, 5 wyvilSi A efeqturi da B ue-feqto iyo, xolo 16-Si B iyo efeqturi da A uefeqto.

3.1. romeli kriteriumi unda gamoviyenoT wamlebis efeqturobis Sesa-

dareblad?

3.2. gamoTvaleT kriteriumis p-mniSvneloba.

4. Catarda igive gamokvleva, rac 3 amocanaSi mxolod kacebSi

da 100 dawyvilebuli monacemidan aRmoCnda, rom 52-ze orive wamalma

dadebiTad imoqmeda, 35-ze arcerTma wamalma ar iqonia gavlena, 5 wyvi-lSi A efeqturi da B uefeqto iyo, xolo 16-Si B iyo efeqturi daA uefeqto.

4.1. ramdeni konkordantuli wyvilia mamakacebSi?4.2. ramdenia diskordantuli wyvili?

4.3. aageT wamlebis efeqturobis Sesadarebeli kriteriumi da gamoTva-

leT kriteriumis p-mniSvneloba.

l e q c i a 14.

Tavi 13. regresiuli analizi da korelacia.

13.1. Sesavali, ZiriTadi cnebebi.wina leqciebSi Cven SeviswvleT statistikuri meTodebi SerCev-

iT monacemebze dayrdnobiT ori an meti normaluri populaciis saSua-

loebis Sesadareblad (leqciebi 9 da 10), Semdeg daaxloebiT igive

gavakeTeT kategorizebuli monacemebisaTvis (leqciebi 11 da 12). Cven

davinaxavT, rom es meTodebi Sesabamisad, warmoadgens e.w. wrfivi da

logisturi regresiis meTodebis kerZo SemTxvevebs, sadac kvlav lapa-

rakia kavSirebze or an met SemTxveviT sidides Soris. kerZod, am naw-

ilSi, rogorc wesi, Cven erTis mxriv, gvainteresebs urTierTkavSiriSesaswavl sidideebs Soris, xolo meores mxriv, gvainteresebs erT-

erTi (an ramdenime) maTganis damokidebuleba danarCen sidideebze. pirv-el SemTxvevaSi Cven dagvWirdeba korelaciis Teoria, xolo meoreSi

regresiuli analizi. magaliTad, rodesac Cven vlaparakobT adamianis

simaRlisa da wonis urTierTkavSiris Sesaxeb, maSin saqme gvaqvs kore-

laciuri analizis amocanasTan, xolo roca Cven gvinda simaRlis mixe-

dviT daskvna gavakeToT wonis Sesaxeb, saubaria ukve regresiuli anal-

izis amocanaze.

magaliTi 13.1. sameano saqmeSi atareben dakvirvebas estriolis

(estriol) doneze orsuli qalis Sardis sinjSi, romelsac male mouwevs

mSobiaroba, radgan dadgenilia, rom estriolis done kavSirSia axalS-

obilis wonasTan. es kavSiri SeiZleba gamoisaxos regresiis wrfis sa-

180

SualebiT. Greene –ma da Touchstone–ma Seiswavles aseTi urTierTkavSi-ri da miiRes Semdegi Sedegebi:

iestrioli

mg/24sT, xi

wona g/100,yi

iestrioli

mg/24sT, xi

wona g/100,yi

1 7 25 17 17 322 9 25 18 25 323 9 25 19 27 344 12 27 20 15 345 14 27 21 15 346 16 27 22 15 357 16 24 23 16 358 14 30 24 19 349 16 30 25 18 3510 16 31 26 17 3611 17 30 27 18 3712 19 31 28 20 3813 21 30 29 22 4014 24 28 30 25 3915 15 32 31 24 4316 16 32

rogorc vxedavT, x-ebiT aRniSnulia estriolis done, xolo y-ebiT ki axalSobilis wona. ra Tqma unda, ar SeiZleba saubari raime

funqcionalur kavSirze x da y sidideebs Soris, rogorc es fizikisuniversalur sidideebs Soris kavSiris dros xdeba, radgan rogorc

adre SevTanxmdiT, x da y sidideebs (ricxvebs) Cven vuyurebT, rogorcX da Y SemTxveviTi sidideebis realizaciebs, romelTa Soris kavSir-

ze laparaki SeiZleba mxolod stoqasturi (albaTuri) azriT. amitom

regresiul analizSi ixilaven Y SemTxveviTi sididis pirobiT saSua-lo yofaqcevas, im pirobiT, rom ganxorcielda {X = x} xdomiloba, anuixilaven E(Y |X = x) pirobiT maTematikur lodins, rogorc x cladis

funqcias

y = r(x) E(Y | X = x). (13.1)

am funqcias uwodeben swored, regresiis funqcias (wirs) Y SemTxvevi-Ti sididisa X SemTxveviTi sididiT. magaliTad, Cven SeiZleba gvainte-

resebdes, rogoria 175 sm simaRlis adamianebis saSualo wona, an piri-

qiT, saSualod ra simaRlisa arian 80 kg wonis mqone adamianebi? rog-

orc vxedavT, (13.1)-Si garkvuli azriT, laparakia Y SemTxveviTi sidi-dis prognozirebaze X SemTxveviTi sididiT. amdenad, X SemTxveviTi si-dides (an mis Sesabamis x cvlads) damoukidebel sidides, xolo Y Se-mTxveviTi sidides (an mis Sesabamis y cvlads) damokidebul sidides,(gamoZaxils, an prognozs) uwodeben.

radgan Cven araferi gviTqvams X da Y SemTxveviTi sidideebiserTobliv ganawilebaze (da es tipiurad ase iqneba praqtikaSi), stati-

181

stikaSi da kerZod, regresiul analizSi akeTeben daSvebas regresiisr(x) funqciis saxeze (magaliTad, daSvebas misi wrfivobis, an eqsponen-

cialurobis Sesaxeb da a.S.). es daSvebebi anu e.w. regresiuli modele-bi, rogorc wesi, Seicavs parametrebis garkveul raodenobas da maSin

regresiuli analizis amocanaa parametrebis SerCevis xarjze monaceme-bis Sesaferisi modelis morgeba.

am nawilSi Cven upiratesad saubari gveqneba Semdeg wrfiv regr-

esiul modelze:

y = r(x) = a + bx, (13.2)sadac a da b modelis parametrebia.

imisaTvis, rom sworad SeirCes regresiuli modeli, SerCeviT

monacemebs warmoadgenen sibrtyeze grafikulad, rasac gafantulobisdiagramas uwodeben. mis asagebad saWiroa dakvirvebuli ricxviTi wyvi-

lebi (x1,y1), (x2,y2), …, (xn,yn) davitanoT sakoordinato xOy sibrtyeze.

Cveni magaliTisaTvis gafantulobis diagrama aseTia:

05

101520253035404550

0 5 10 15 20 25 30

estrioli

wona

rogorc vxedavT, wertilebi marTlac “mimofantulia” sibrtye-

ze. regresiis wrfis agebamde (a da b koeficientebis gansazRvramde),SevniSnoT, rom y = a + bx Tanadoba zustad albaT ar Sesruldeba yve-

la monacemisaTvis, anu yi = a + bxi toloba ar aris samarTliani yove-li i-saTvis da maSasadame, damoukidebeli sididis yovel mocemul xi

mniSvnelobas (esteriolis dones Cveni magaliTisaTvis), Tan axlavs

garkveuli ei Secdoma (naSTi) yi-saTvis, romelic ase SeiZleba gamoisa-

xos:

ei = yi –(a + bxi). (13.3)

am Secdomis arseboba Cvens magaliTSi SeiZleba aixsnas axalSob-

ilTa wonebis variaciiT dedebis estriolis doneebis tolobisas. ami-tom sruli wrfivi regresiuli modeli ase gamoiyureba:

182

Y = a + bX + E, (13.4)sadac e-s Sesabamisi E SemTxveviTi sididisaTvis Cven vuSvebT, rom is

normaluradaa ganawilebuli saSualoTi 0 da dispersiiT 2.

13.2. umcires kvadratTa meTodi.

davubrundeT (x1,y1), (x2,y2), …, (xn,yn) dakvirvebebis saSualebiT

regresiis wrfis a da b koeficientebis (parametrebis) gansazRvris sa-kiTxs. zogad meTodologias, romlis saSualebiTac akeTeben amas, umc-ires kvadratTa meTodi hqvia, romlis arsic SeiZleba Semdegnairad av-

xsnaT: radgan gafantulobis diagramaze SeiniSneba (x1,y1), (x2,y2), …,(xn,yn) dakvirvebebis wrfivad dalagebis tendencia, amitom Cveni amoca-

naa iseTi wrfis ageba, romelic “axlos gaivlis” dakvirvebis yvela

wertilTan, an ufro zustad, yvela wertilis jamuri daSoreba (OyRerZis gaswvriv) am wrfidan iqneba minimaluri. magram es daSoreba i-uri wertilisaTvis gamoisaxeba swored (13.3)-iT gansazRvruli ei-uri

Secdomis absoluturi sididiT: | ei | = | yi –(a + bxi)|. maSasadame, Cveniamocanaa movZebnoT iseTi a da b koeficientebi, rom minimaluri iyos

n

iii

n

ii bxayebaSAE

11

|)(|||),( gamosaxulebis mniSvneloba, magram

rogorc adre dispersiis gansazRvrisas, moxerxebulia visaubroT araSecdomebis absoluturi sidideebis, aramed maTi kvadratebis jamze da

ganvixiloT aseTi (ori cvladis funqciis) minimizaciis sakiTxi:

n

iii

n

ii bxayebaSSE

1

2

1

2 )(),( . (13.5)

ori cvladis funqciis eqstremumis pirobebidan gamodis, rom

(13.5) gamosaxuleba minimums aRwevs a da b parametrebis iseT a~ da b~

mniSvnelobebisaTvis, romelTaTvisac sruldeba pirobebi: b~

= Lxy / Lxx da

nxxxynnn xLLyxbya )/(~~ , sadac ny da nx warmoadgens Sesaba-

misad, y-ebisa da x-ebis SerCeviT saSualoebs da

n

ininixy yyxxL

1

)()( , (13.6)

n

ini

n

ininixx xxxxxxL

1

2

1

)()()( . (13.7)

sabolood, Tu a da b parametrebis a~ da b~ mniSvnelobebs CavsvamT

regresiis wrfis (13.2) gantolebaSi, miviRebT:

y - ny = (Lxy / Lxx)(x - nx ). (13.8)

davubrundeT Cvens magaliTs da vnaxoT, rogor saxes miiRebs

regresiis wrfis gantoleba. am SemTxvevaSi:

183

ny = (1/31)(25 + 25 + 25 + 27 +…+ 40 + 39 + 43) = 992/31 = 32;

nx = (1/31)(7 + 9 + 9 + 12 +…+ 22 + 25 + 24) = 534/31 = 17.23;Lxy = (25-32)(7-17.23) + (25-32)(9-17.23) + (25-32)(9-17.23)+

+ (27-32)(12-17.23) +…+ (40-32)(22-17.23) + (39-32)(25-17.23) ++ (43-32)(24-17.23) = 412;

Lxx = (7-17.23)2 + (7-17.23)2+(7-17.23)2+(7-17.23)2+…++(7-17.23)2+(7-17.23)2+(7-17.23)2 = 677.42.

amitom (13.8)-is mixedviT, regresiis wrfis gantolebaa

y – 32 = (412 /677.42)(x –17.23) = 0.608(x –17.23),an gamartivebis Semdeg miviRebT: y = 21.525 + 0.608x.

13.3. regresiis wrfis parametrebis aRricxva.

SevniSnoT, rom (13.8)-iT gansazRvruli regresiis wrfe yovelT-

vis gadis wertilze koordinatebiT ( nx , ny ), xolo misi sakuTxo koe-

ficientis mniSvnelSi mdgomi Lxx sidide warmoadgens (n-1)-jer x-ebisSerCeviT dispersias, anu Lxx = (n - 1) 2

ns . garda amisa, imave koeficient-

is mricxvelSi mdgomi sidide, romelic Cven momavalSic xSirad Segvx-

vdeba, warmoadgens x-ebisa da y-ebis maTi SerCeviTi saSualoebisagan

gadaxrebis namravlebis jams da Lxy / (n - 1) SeiZleba gaviazroT, rog-

orc am gadaxrebis SerCeviTi saSualo. am maxasiaTebels x-isa da y-is(ufro swored, X-isa da Y-is) SerCeviTi kovariaciis koeficients uw-

odeben. misi Sinaarsi imaSi mdgomareobs, rom is gviCvenebs saSualod

X da Y ixrebian TavianTi saSualoebidan erTsadaimave, Tu sxvadasxvamxares. am koeficients ufro dawvrilebiT Cven qvemoT SevexebiT.

rogorc wina punqtSi vTqviT, regresiis wrfe ver gaivlis yve-

la SerCeviT wertilze. magram yoveli xi-s regresiis wrfeze Seesaba-

meba wertili, romlis ordinati aRvniSnoT iy~ –iT. maSasadame, iy~

a~ +b~xi. Cven gvinda yuradReba gavamaxviloT sam sxvadasxvanair sxvaob-

aze, romelic dakavSirebulia yi-Tan, anu gamoZaxilis SerCeviT mniSvne-

lobasTan: ny -Tan, yi-is SerCeviT saSualosTan da xi-is Sesabamisi reg-

resiis wrfis wertilis iy~ ordinatTan. es sxvaobebia: yi - ny , iy~ - ny

da yi - iy~ . cxadia, rom maT Soris aseTi kavSiria:

yi - ny = ( iy~ - ny ) + (yi - iy~ ). (13.9)

maSasadame, SerCeviTi mniSvnelobis gadaxra Tavisi saSualodanwarmodgeba ori danarCeni sxvaobis (komponentis) jamis saxiT: pirvels

uwodeben regresiis komponents, xolo meores – naSTiT komponents(wevrs). grafikulad es sidideebi Semdegnairad SegviZlia warmovadgi-

noT:

184

naxazidan Cans, rom Tu SerCeviTi wertili moxvdeba regresiis

wrfeze, maSin yi = iy~ , anu yi - iy~ = 0 da e.i. naSTiTi wevri 0-is tolia.

sxvanairad rom vTqvaT, kargad morgebuli regresiis wrfis SemTxveva-

Si regresiis komponenti ufro didia naSTiT wevrze da piriqiT, Tu

naSTiTi wevri ufro didia, vidre regresiis komponenti, unda vifiqr-

oT, rom regresiis wrfe cudadaa SerCeuli. magram aq isev laparakia

yovel calkeul wertilze da Cven ki gvinda davaxasiaToT gadaxrissidideebi yvela wertilisaTvis mTlianobaSi. amitom ganvixiloT (ro-

gorc es gavakeTeT dispersiul analizSi) yi - ny gadaxrebis kvadrate-

bis jami, anu e.w. sruli (mTeli) kvadratebis jami (Total sum of squar-es), romelsac Cven TSS simboloTi aRvniSnavT:

TSS =

n

ini yy

1

2)( . (13.10)

maSin (13.9)-dan dispersiuli analizis SemTxvevis msgavsad, Seg-viZlia es jami davSaloT or jamad TSS = RSS + ESS , sadac

RSS =

n

ini yy

1

2)~( da ESS =

n

iii yy

1

2)~( , (13.11)

anu RSS warmoadgens regresiis komponentTa kvadratebis jams (Regress-ion sum of squares), xolo – ESS naSTiT wevrTa kvadratebis jams (Er-ror (Residual) sum of squares). SevniSnoT, rom es sidideebi maTi agebiswesidan gamomdinare, igive rols TamaSobs regresiul analizSi, rogo-

rsac jgufTaSorisi BSS da Sidajgufuri WSS variaciebi dispersiulanalizSi.

es sidideebi SeiZleba gamovsaxoT (13.6) da (13.7)-iT ganmarteb-

uli sidideebiT Semdegnairad:

RSS = xxxyxxxy LLLbLb /~~ 22 da ESS = TSS - RSS = Lyy - xxxy LL /2 . (13.12)

185

13.4. Tanxmobis kriteriumebi regresiis wrfisaTvis.13.4.1. Tanxmobis F–kriteriumi.

dispersiul analizTan zemoaRniSnuli msgavsebidan gamomdinare

Cven SegviZlia avagoT modelisa da monacemebis Tanxmobis kriteriumi.

kerZod, Tu ganvixilavT RSS da ESS sidideebis Sefardebas da aRmoCn-deba, rom es Sefardeba didia, maSin unda vifiqroT, rom monacemebieTanxmeba models, xolo am fardobis SemTxvevaSi, Tanxmoba “cudia”.

hipoTezaTa Semowmebis amocanis terminebSi es ase gamoiTqmeba: nulova-

ni hipoTezaa, rom H0 : b = 0, xolo alternativaa H1 : b 0, sadac baris (13.4) modelSi wrfis sakuTxo koeficienti. kriteriumis asageb-

ad SemovitanoT Semdegi sidideebi, RMS RSS / k da EMS ESS /(n- k-1),sadac k aRniSnavs damoukidebeli sidideebis raodenobas (Cvens mier

ganxilul SemTxvevaSi, k =1) da ganvixiloT Semdegi statistika:

T = RMS / EMS = [( xxxy LL /2 ) / k]/[(Lyy - xxxy LL /2 )/(n-k -1)]. (13.13)

mtkicdeba, rom H0 hipoTezis samarTlianobis SemTxvevaSi am statisti-

kas aqvs F(k, n-k-1)-ganawileba da maSasadame, statistikuri kriteriumi

aseTnairad yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis T statistikisdakvirvebuli t mniSvneloba akmayofilebs pirobas t > Fk,n-k-1,1-, maSin mniSvnelovnebis doniT H0 hipoTezas uarvyofT, winaaRmdeg SemTxvevaSi

amis safuZveli ara gvaqvs.

kriteriumis Sesabamisi p-mniSvneloba tolia p = P{ Fk , n-k-1 > t}.kriteriumis gamoyenebis sailustraciod davubrundeT Cvens mag-

aliTs da vnaxoT, ramdenad eTanxmeba monacemebi wrfiv regresiul mo-dels. amisaTvis gavixsenoT, rom Lxy = 412, Lxx = 677.42.

garda amisa, Lyy = 674 (SeamowmeT!) da amitom (13.12) formuli-

dan gveqneba: RSS = 4122/677.42 = 250.57; TSS = Lyy = 674; ESS = 674 - 250.57 = 423.43;

RMS = 250.57/1 = 250.57; EMS = 423.23/(31-1-1) = 14.60.bolos, (13.13) formulidan gvaqvs:

t = 250.57 / 14.60 = 17.16.F-ganawilebis kvantilebis cxrilidan vpoulobT, rom

Fk,n-k-1,0.999 = F1, 29, 0.999 = 14.82.radgan F1, 29, 0.999 = 14.82 < 17.16 = t, amitom = 0.001 mniSvnelovne-

bis doniT H0 hipoTezas uarvyofT da miviRebT alternativas, rom es-

triolis donesa da axalSobilis wonas Soris arsebobs wrfivi kavSi-

ri.

xazi gavusvaT kidev erT sidides (koeficients). Ees aris e.w.

determinaciis koeficienti, romelic aRiniSneba R2-iT da ganisazRvre-

186

ba, rogorc regresiis komponentTa kvadratebis jamis fardoba, kvadra-tebis mTlian jamTan, anu

R2 = RSS / TSS = )/(2yyxxxy LLL . (13.14)

am koeficientis azri SeiZleba ase gavigoT: Tu R2 = 1, maSin(13.4) modelSi sruli variacia modis damoukidebel X cvladze (sid-

ideze), xolo Tu R2 = 0, maSin sruli variacia modis SemTxveviT E ko-

mponentze, anu X SemTxveviTi sidide gavlenas ar axdens gamoZaxilze

da maSasadame, rac ufro didia am koeficientis mniSvneloba, YSemTxveviTi sididis variacia miT ufroa gansazRvruli X SemTxveviTisididiT.

Cveni magaliTis SemTxvevaSi, R2 = 250.57 / 674 = 0.372 da maSasada-me, axalSobilTa wonebis variaciis 37.2% modis estriolis doneze.

SevniSnoT, rom roca n didia, EMS = (Lyy - xxxy LL /2 )/(n-k -1) =

= Lyy (1-R2) /(n-k -1) = 2ys (1-R2)[(n-1) /(n-k -1)] 2

ys (1-R2),

saidanac

1 - R2 EMS / 2ys = 2

|xys / 2ys , (13.15)

sadac 2|xys -iT aRniSnulia Y -is pirobiTi dispersiis “saukeTeso” Sefa-

seba pirobaSi {X = x}.maSasadame, (13.15) toloba gveubneba, rom 1-R2 warmoadgens Y -is

dispersiis im proporcias, romelic SeiZleba aixsnas X -iT.

Cveni magaliTisaTvis 2|xys / 2

ys 1 - R2 = 1- 0.372 = 0.628.

13.4.2. Tanxmobis t-kriteriumi.igive amocanis gadasaWrelad, H0 : b = 0 hipoTezis Sesamowmeblad

H1 : b 0 alternativis winaaRmdeg, iyeneben sxva kriteriumsac, rome-

lic damyarebulia (13.6)-iTa da (13.7)-iT gansazRvrul b~

= Lxy / Lxx ko-

eficientze, romelsac Tu ganvixilavT rogorc SemTxveviT sidides,

hipoTezis samarTlianobisas aqvs 0-is toli saSualo da 2 / Lxx-is

toli dispersia. radgan sazogadod, 2 ucnobia, mas afaseben sididiT2|xys . ganvixiloT statistika:

T = b~

/ ( 2|xys /Lxx)

1/2, (13.16)

romelsac mtkicdeba, rom H0 hipoTezis samarTlianobis dros aqvs

t(n-2)-ganawileba. amitom statistikuri kriteriumi ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis T statistikisdakvirvebuli t mniSvneloba akmayofilebs pirobas

-tn – 2 , 1 - /2 t tn – 2 , 1 - /2,

187

maSin mniSvnelovnebis doniT H0 hipoTezis uaryofis safuZveli aragvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. cxadia, rom kriteriumis

Sesabamisi p-mniSvneloba gamoiTvleba Semdegnairad:

p = 2 P{ tn – 2 < t}, roca t < 0 da p = 2 P{ tn – 2 > t}, roca t > 0.kriteriumis gamoyenebis sailustraciod davubrundeT 13.1 maga-

liTs da vnaxoT, xom ar iZleva es kriteriumi sxva Sedegs, vidre uk-ve ganxiluli F-kriteriumi.

rogorc gvaxsovs, b~

= Lxy / Lxx = 0.608. amitom T statistikis dak-

virvebuli t mniSvneloba tolia

t = b~

/ ( 2|xys /Lxx)

1/2 = 0.608/(14.60/677.42)1/2 = 0.608/0.147 = 4.14.

t-ganawilebis kvantilebis cxrilidan vpoulobT, rom t29, 0.9995 =3.659. radgan t2 9, 0.9995 = 3.659 < 4.14 = t, amitom es kriteriumic uary-ofs H0 hipoTezas. unda iTqvas, rom es orive kriteriumi sazogadod-

ac erTnair Sedegebs iZleva, anu isini eqvivalenturia, magram ukanaskn-

eli maTgani, xSirad gamoiyeneba ndobis intervalis agebisas b paramet-

risaTvis. am sakiTxs Cven swored momdevno punqtSi ganvixilavT.

13.5. intervaluri Sefasebebi wrfivi regresiisaTvis.

13.5.1. ndobis intervalebi regresiis parametrebisaTvis.

regresiis wrfis parametrebis standartuli Secdomebi da maTi

intervaluri Sefasebebi xSirad gamoiyeneba imisaTvis, rom davadginoT

aris Tu ara Sedarebadi Cvens mier miRebuli Sedegebi sxva msgavsi gam-

okvlevebiT miRebul SedegebTan.

magaliTi 13.2. davuSvaT, adrindeli dakvirvebebidan 500 orsulqalze cnobilia, rom regresiis wrfis gantolebas aqvs Semdegi saxe:

y = 25.04 + 0.52x. ra SeiZleba iTqvas Cveni Sedegebis Sesaxeb 13.1 maga-liTSi?

rogorc wina punqtSi vnaxeT, b~ parametris standartuli Sec-

doma gamoiTvleboda formuliT xxxy Lsbse /)~

( 2| . Tu kvlav gavixsen-

ebT (13.6)-sa da (13.7) formulebs, a~ parametris standartuli Secdo-

misaTvis miviRebT:

)//1()~( 22| xxnxy Lxnsase . (13.17)

aseve wina punqtidan Cven viciT, rom T = b~

/ ( 2|xys /Lxx)

1/2 statist-

ikas H0 hipoTezis samarTlianobis dros hqonda t(n-2)-ganawileba. amit-

om 100(1-)%-ian ndobis intervalebs b da a parametrebisaTvis Sesab-amisad aqvs Semdegi saxe:

b~ tn – 2 , 1- /2 xxxy Ls /2

| da a~ tn – 2 , 1- /2 )//1( 22| xxnxy Lxns . (13.18)

188

mivubrundeT 13.2 magaliTs da SevecadoT pasuxi gavceT dasmulSekiTxvas. amisaTvis vnaxoT rogori saxe aqvs 95%-ian ndobis interva-

lebs b da a parametrebisaTvis 13.1 magaliTSi. rogorc gvaxsovs,

xxxy Ls /2| = 0.147 da t -ganawilebis kvantilebis crilidan t29, 0.975 =

2.045. amitom (13.18)-dan gveqneba:

b~ tn – 2 , 1- /2 xxxy Ls /2

| = 0.608 2.0450.147 = (0.308; 0.908).

analogiurad, )//1( 22| xxnxy Lxns =[14.60(1/31+17.232/677.42)]1/2 = 2.62

da

a~ tn – 2 , 1- /2 )//1( 22| xxnxy Lxns = 21.525 2.0452.62 = (16.16, 26.88).

rogorc vxedavT, intervalebi sakmarisad farTea, rac cxadia,

gamowveulia SerCevis moculobis simciriT. miuxedavad amisa, b da aparametrebis mniSvnelobebi vardeba Cvens mier agebul ndobis interva-

lebSi da SegviZlia davaskvnaT, rom magaliT 13.1-Si agebuli regresiis

gantoleba Sedarebadia magaliT 13.2-Si moyvanil SedegTan.

13.5.2. ndobis intervalebi regresiis prognozebisaTvis.

rogorc adrec aRvniSneT, regresiis wrfis mniSvnelovan gamoye-

nebiT mxares warmoadgens misi gamoyeneba prognozebis misaRebad. axla

Cven gvainteresebs sakiTxi, Tu ramdenad “kargia” miRebuli prognoze-

bi? 13.1 magaliTis terminebSi, Tu ganvixilavT estriolis fiqsirebul

x donian yvela orsul qals, maSin regresiis wrfidan miRebuli axal-

Sobilis wonis prognozia y~ = a~ +b~ x da maSasadame, Cveni SekiTxva as-

eTia: ramdenad “kargia” axalSobilis wonis es prognozi? am kiTxvaze

pasuxi damokidebulia imaze, Tu ras vaprognozebT Cven: estriolis xdoniani konkretuli orsuli qalis, Tu estriolis x doniani yvela

orsuli qalis axalSobilis wonas. amis mixedviT, pirvel SemTxvevaSi,

Y SemTxveviTi sidide ganawilebulia normalurad, saSualoTi y~ da

Sefasebis standartuli SecdomiT,

se1 = )/)(/11( 22| xxnxy Lxxns , (13.19)

xolo Sesabamis 100(1-)%-ian ndobis intervals aqvs Semdegi saxe:

y~ tn – 2 , 1- /2 )/)(/11( 22| xxnxy Lxxns . (13.20)

meore SemTxvevaSic, Y SemTxveviTi sididis saSualos saukeTeso

Sefasebaa y~ da misi standartuli Secdoma moicema formuliT

se2 = )/)(/1( 22| xxnxy Lxxns , (13.21)

xolo Sesabamis 100(1-)%-ian ndobis intervals aqvs Semdegi saxe:

189

y~ tn – 2 , 1- /2 )/)(/1( 22| xxnxy Lxxns . (13.22)

Tu davakvirdebiT (13.19)-(13.22) formulebs, davinaxavT, romorive SemTxvevaSi standartuli Secdoma arsebiTadaa damokidebuli im-

aze, Tu ramdenad axlosaa saintereso x sidide dakvirvebaTa xi SerCev-

iT saSualosTan: rac ufro axlosaa es sidideebi erTmaneTTan, miT

ufro mcirea Secdomebi da maSasadame, miT ufro viwroa ndobis inte-

rvalebi da piriqiT, rac ufro didia am sidideebs Soris daSoreba,

miT ufro didia Secdomebi da maSasadame, miT ufro farTea ndobis in-

tervalebi. garda amisa, SevniSnoT, rom meore SemTxvevaSi ndobis inte-

rvali ufro viwroa, vidre pirvelSi, rac gamowveulia imave efeqtiT,

rom SemTxveviTi sididis SerCeviTi saSualos standartuli Secdoma

miT ufro pataraa, rac ufro meti SerCeviTi monacemiT xdeba misi Se-faseba.

am formulebis gamoyenebis sailustraciod ganvixiloT 13.1 mag-

aliTis Semdegi modifikaciebi.

magaliTi 13.3. davuSvaT, Cven gvainteresebs aris Tu ara normaSi

vinme, elizabet jonsonis axalSobilis wona, 1850 grami, Tu misi est-

riolis done mSobiarobis win Seadgenda 10 mg/24sT-s?

magaliTi 13.4. avagoT 95%-iani ndobis intervali axalSobilis

wonis saSualosaTvis, im dedebisaTvis, romelTa estriolis done mSo-

biarobis win Seadgenda 10 mg/24sT-s.

magaliTi 13.3-is amoxsna. gavixsenoT regresiis wrfis gantole-

ba 13.1 magaliTidan: y = 21.525 + 0.608x da vnaxoT, rogoria axalSobi-

lis wonis mniSvneloba x = 10-saTvis:

y = 21.525 + 0.608 10 = 21.525 + 6.08 = 27. 605 = 2760(grami).

gamovTvaloT Sesabamisi standartuli Secdoma. gavixsenoT, rom

nx = 17.23, Lxx = 677.42 da 2|xys = 14.60. amitom (13.19) formulidan mivi-

RebT: se1 = )/)(/11( 22| xxnxy Lxxns =

= [14.60(1+1/31+(10-17.23)2/677.42)]1/2 = 4.025.

Sesabamisad, 95%-ian ndobis intervals axalSobilis saSualo

wonisaTvis aqvs Semdegi saxe:

27.6052.0454.025=27.6058.231=(19.434, 35.836)=(1943g, 3583g).

radgan magaliTSi aRebuli 1850 grami ar vardeba miRebul inte-

rvalSi, davaskvniT, rom elizabet jonsonis axalSobilis wona ar Se-

esabameba normas.

190

magaliTi 13.4-is amoxsna. kvlav, radgan nx = 17.23, Lxx = 677.42

da 2|xys = 14.60 da y = 21.525 + 0.608 10 = 27. 605 = 2760(grami),

amitom (13.21) formuliT gamovTvaloT standartuli Secdoma x = 10-

saTvis. gvaqvs: se2 = )/)(/1( 22| xxnxy Lxxns =

= [14.60(1/31+(10-17.23)2/677.42)]1/2 = 1.264.

Sesabamisad, 95%-ian ndobis intervals axalSobilis saSualo

wonisaTvis aqvs Semdegi saxe:

27.6052.0451.264=27.6052.585=(25.02, 30.190)=(2502g, 3019g).

13.6. naSTTa analizi martivi wrfivi regresiisaTvis.

gavixsenoT (13.4) sruli wrfivi regresiuli modeli. rogorc

gvaxsovs, Y = a + bX + E, sadac E SemTxveviTi sidide, romelsac Cven

naSTi vuwodeT, normalurad iyo ganawilebuli saSualoTi 0 da dispe-rsiiT 2. Sesabamisi i-uri naSTis SerCeviTi mniSvneloba ganisazRvreb-

oda rogorc sxvaoba gamoZaxilis SerCeviT mniSvnelobasa da regresiis

wrfis mniSvnelobas Soris xi wertilSi, anu ei = yi - iy~ , sadac

iy~ = a~ + b~xi. am nawilSi gansaxilveli sakiTxi gulisxmobs swored ei, i

=1,2,…,n, naSTebis daxasiaTebas, anu imas Tu ramdenad eTanxmeba es na-STebi modelis daSvebebs. es daSvebebi ki aseTia:

1) yoveli mocemuli x-saTvis Y SemTxveviTi sididis pirobiTi

maTematikuri lodini pirobaSi {X = x} xdomiloba, warmoadgens x-iswrfiv funqcias, anu E(Y | X = x) = a + bx;

2) yoveli mocemuli x-saTvis Y SemTxveviTi sididis pirobiTiganawileba pirobaSi {X = x} xdomiloba, normaluria saSualoTi

a + bx da dispersiiT 2, romelic erTidaigivea yvela x-saTvis;3) Ei, i =1,2,…,n, naSTebi damoukidebelia.

rogorc wesi, am daSvebebis samarTlianobis Sesamowmeblad iyen-

eben sxvadasxva grafikul warmodgenebs. erT-erTi aseTi grafikuli xe-

rxi mdgomareobs grafikze ( iy~ , ei / se(ei)) wertilebis dataniT, sadac

se(ei) warmoadgens naSTis standartul Secdomas, romelic ase gamoiT-

vleba:

se(ei) = )/)(/11( 22| xxnxy Lxxns . (13.23)

maSin grafikuli kriteriumi mdgomareobs SemdegSi: Tu wertil-

Ta miRebuli erToblioba dalagdeba Tanabrad Ox RerZis gaswvriv (ze-moT da qvemoT), vTvliT, rom yvela daSveba met-naklebad sruldeba,

191

winaaRmdeg SemTxvevaSi, unda vifiqroT, rom dairRva romeliRac piro-ba. leqciis bolos movyavs aseTi diagramis saxe 13.1 magaliTisaTvis:

rogorc vxedavT, ( iy~ , ei / se(ei)) wertilebi y = 0 wrfis gaswvriv

dalagda. maSasadame, Cven vTvliT, rom Sesrulebulia wrfivi regresi-

uli modelis yvela piroba.

13.7. korelaciis koeficienti.

13.7.1. erTamokrefiani kriteriumebi korelaciis

koeficientisaTvis.

aqamde Cven vixilavdiT regresiuli analizis amocanebs, sadac

gvainteresebda damokidebuli cvladis prognozirebis sakiTxi erTi an

ramdenime damoukidebeli cvladis saSualebiT. magram xSirad gvainter-

esebs ara romelime sididis prognozireba, aramed ubralod, aris Tuara kavSiri dakvirvebad sidideebs Soris. aseT SemTxvevebSi iyeneben

korelaciis koeficients or sidides Soris, romelic r-iT aRiniSnebada me-13 leqciaSi Semotanili sidideebis saSualebiT ase gansazRvreba:

r =yx

xy

yyxx

xy

ss

nL

LL

L

)1/(. (13.24)

sadac sx da sy warmoadgens Sesabamisad x-ebisa da y-ebis SerCeviT sta-ndartul gadaxrebs da (13.7)-iT ganmartebuli Lxy sidide warmoadgens

192

x-ebisa da y-ebis Sereul centralur SerCeviT moments da Lxy/(n-1) si-dides x-ebisa da y-ebis SerCeviT kovariaciis koeficients uwodeben.

gavixsenoT (13.14) Tanafardobidan determinaciis koeficientis

gansazRvreba. rogorc gvaxsovs, R2 = )/(2yyxxxy LLL da maSasadame,

(13.24) tolobidan davaskvniT, rom R2 = r2, anu korelaciis koeficien-

tis kvadrati emTxveva determinaciis koeficients. amitom erTi Sexed-

viT, korelaciis koeficientis SemotanaSi axali araferi unda iyos,

magram determinaciis koeficientisagan gansxvavebiT misi niSani gviCv-

enebs, Tu rogoria kavSiri or sidides Soris, dadebiTi Tu uaryofiTianu is gviCvenebs X da Y SemTxveviTi sidideebi dadebiTadaa korelir-ebuli Tu uaryofiTad. dadebiTad korelirebuloba niSnavs imas, romX-is mier didi da patara mniSvnelobebis miReba iwvevs Y-is SesabamisimniSvnelobebis miRebas. piriqiT, uaryofiTad korelirebuloba ki imas

niSnavs, rom X-is did mniSvnelobebs mohyveba Y-is patara mniSvnelobe-

bi, xolo pataras – didi.

garda amisa, Tu | r | 1, niSnavs Zlier (TiTqmis wrfiv) kavSirs

Sesabamis sidideebs Soris, xolo | r | 0, niSnavs sakmarisad sust kav-Sirs mocemul sidideebs Soris.

gavixsenoT axalSobilTa wonebisa da estriolis donis Tanado-

bis 13.1 magaliTi. rogorc gvaxsovs, Lxy = 412, Lxx = 677.42, Lyy = 674. am-itom (13.24) formulidan miviRebT, rom r = 412/(677.42674)1/2 = 0.61.

sainteresoa agreTve kavSiri korelaciisa da regresiis koefic-

ientebs Soris martiv wrfiv regresiis modelSi. rogorc gvaxsovs,

b~

= Lxy / Lxx. martivi gardaqmnebiT davaskvniT, rom

r = b~(sy/sx) (13.25)

da kvlav TiTqos regresiis koeficientisaTvis moxda mxolod skalis,

masStabis Secvla. magram Tu kargad davakvirdebiT, aRmovaCenT, rom b~-

sagan gansxvavebiT korelaciis koeficients ar gaaCnia ganzomileba, anu

is ganyenebuli ricxvia. es kargad Cans isev 13.1 magaliTidan. am magal-

iTSi b~koeficients aqvs aseTi ganzomileba (grami/100)/(mg/24sT), rac

cota ar iyos gaugebari erTeulia TavisTavad. maSin, roca korelaciis

koeficients es nakli ar gaaCnia.imisaTvis, rom vawarmooT statistikuri daskvnebi korelaciis

SerCeviTi koeficientis saSualebiT, saWiroa msgavsi cneba gvqondes

populaciebisaTvisac, anu X da Y SemTxveviTi sidideebisaTvisac.

(13.24) formula warmoadgens SerCeviT analogs Semdegi gamosaxulebi-

saTvis

YX

EYYEXXE

)()(, (13.26)

193

romelsac X da Y SemTxveviTi sidideebis korelaciis koeficienti ew-odeba.

korelaciuri analizis erT-erTi ZiriTadi amocana imaSi mdgoma-

reobs, rom monacemebze dayrdnobiT (anu X da Y SemTxveviTi sidideeb-is realizebul dakvirvebebze dayrdnobiT) Semowmdes nulovani hipoT-

eza H0 : = 0 alternativis winaaRmdeg, rom H1 : 0. am amocanis gad-asawyvetad SemoaqvT statistika

21

2

r

nrT

, (13.27)

romelsac H0 hipoTezis samarTlianobis SemTxvevaSi aqvs t(n-2)-ganawi-leba. amitom statistikuri kriteriumi ase yalibdeba:


-tn-2, 1-/2 t tn-2, 1-/2,

maSin mniSvnelovnebis doniT H0 hipoTezis uaryofis safuZveli ara

gvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. kriteriumis Sesabamisi

p-mniSvneloba gamoiTvleba Semdegnairad:

p = 2 P{ tn-2 t}, roca t 0 da p = 2 P{ tn-2 > t}, roca t > 0.cxadia, aq vgulisxmobT, rom X da Y SemTxveviTi sidideebi gan-

awilebulia normalurad.

kriteriumis gamoyenebis sailustraciod ganvixiloT Semdegi

magaliTi 13.5. gulsisxlZarRvTa daavadebis etiologiis erT-

erT mniSvnelovan riskis faqtorad iTvleba SC (Serum Cholesterol)-isdone. bevri gamokvleva mieZRvna garemo faqtor.ebis zemoqmedebas SC-isdonis awevaze. am mizniT gazomes 100 genetikurad daukavSirebuli

col-qmris SC-is doneebi da aRmoCnda, rom SerCeviTi korelaciis

koeficienti tolia r = 0.25. gvaqvs Tu ara safuZveli vamtkicoT, rom

= 0?amoxsna. T statistikis dakvirvebuli t mniSvneloba (13.27)-Si

tolia t = 0.25[98/(1-0.252)]1/2 = 2.56. t–ganawilebis cxrilebSi ver vip-

oviT 98-is Sesabamis kvantilebs. amitom vpoulobT t60, 0.99 = 2.39, t60, 0.995

= 2.66 da t120, 0.99 = 2.358, t120, 0.995 = 2.617, saidanac davaskvniT, rom radg-an 60 < 98 < 120, amitom 0.005 < p/2 < 0.01, anu 0.01 < p < 0.02. sabolo-

od, H0 hipoTezas uarvyofT da miviRebT alternativas, rom 0 damaSasadame, miuxedavad genetikuri kavSirisa, col-qmris qolesterinis

doneebs Soris arsebobs damokidebuleba, rac SeiZleba aixsnas maTi

erTnairi cxovrebis wesiT.

zogjer saWiroa Semowmdes ara H0 : = 0, aramed H0 : = 0 hip-

oTeza, 0-is garkveuli dasaxelebuli mniSvnelobisaTvis, Sesabamisi

alternativis, H1 : 0 dros. garkveuli mizezebis gamo, wina amoca-

194

nis gadasaWrelad gamoyenebuli t-kriteriumi aq aRar gamodgeba. am miz-niT fiSerma SemoiRo korelaciis koeficientis e.w. z-gardaqmna

z = (1/2)(ln(1+ r) - ln(1- r)), (13.28)

romelsac H0 : = 0 hipoTezis samarTlianobis dros aqvs normaluri

ganawileba, saSualoTi z0 = (1/2)(ln(1+ 0) - ln(1- 0)) da dispersiiT1/(n - 3).

Sesabamisad, Tu kriteriumis statistikad aviRebT Semdeg stat-

istikas

T = 3)( 0 nzz , (13.29)

maSin cxadia, mas eqneba standartuli normaluri ganawileba da Sesab-

amisi statistikuri kriteriumi ase Camoyalibdeba:


-z1-/2 t z1-/2,

maSin mniSvnelovnebis doniT H0 hipoTezis uaryofis safuZveli aragvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. kriteriumis Sesabamisi


p = 2 (t), roca t 0 da p = 2 (1- (t)), roca t > 0.cxadia, amave T statistikaze dayrdnobiT da im faqtidan, rom

is standartulad normaluradaa ganawilebuli, Cven SegviZlia avagoT

ndobis intervali ucnobi z0-saTvis da maSasadame, 0-savisac. marTlac,

dasaxelebuli mniSvnelovnebis donisaTvis 1- ndobis albaTobis

mqone anu 100(1-)%-ian ndobis intervals z0-saTvis aqvs Semdegi saxe:

[z – z1-/2 3n , z + z1-/2 3n ]. (13.30)

rac Seexeba 100(1-)%-ian ndobis intervals 0 -saTvis, mas ad-vilad miviRebT (13.30)-dan z0 = (1/2)(ln(1+ 0) - ln(1- 0)) Tanafardobis

gaTvaliswinebiT. cxadia, rom 0 =1

10

0

2

2

z

z

e

e da (13.30)-is sazRvrebis Ca-

smiT, 100(1-)%-ian ndobis intervals 0 -saTvis eqneba Semdegi saxe:

[1

11

1

2

2

z

z

e

e,

1

12

2

2

2

z

z

e

e], (13.31)

sadac z1 = z – z1-/2 3n da z2 = z + z1-/2 3n . (13.32)

sailustraciod ganvixiloT Semdegi

magaliTi 13.6. davuSvaT, 100 mamisa (x) da pirveli biWis (y)wonebs Soris SerCeviTi korelaciis koeficienti gamovida 0.38-is to-li. Cveni amocanaa ganvsazRvroT: 1) aris Tu ara samarTliani hipoTeza

imis Sesaxeb, rom 0 = 0.5, rac mosalodnelia genetikuri kavSiris ga-

mo da 2) avagoT 95%-iani ndobis intervali WeSmariti -saTvis.amoxsna. gamovTvaloT pirvel rigSi z-gardaqmnis dakvirvebuli

da hipoTeturi mniSvnelobebi:

195

z = (1/2)(ln(1+ ) - ln(1- )) = (1/2)(ln(1+ 0.38) - ln(1- 0.38)) == (1/2)ln(2.26) 0.4,

z0 = (1/2)(ln(1+ 0) - ln(1- 0)) = (1/2)(ln(1+ 0.5) - ln(1- 0.5)) == (1/2)ln(3) 0.549.

gamovTvaloT axla T statistikis dakvirvebuli t mniSvneloba:

t = 3)( 0 nzz = (0.4 – 0.549)971/2 -1.47.

maSin kriteriumis p-mniSvneloba tolia

p = 2 (t) = 2 (-1.47) = 2 (1- (1.47)) = 0.142,saidanac, vinaidan p > 0.05, davaskvniT, rom Sedegi statistikurad umn-iSvneloa da maSasadame, H0 : = 0.5 hipoTezis uaryofis safuZveli ara

gvaqvs.

gamovTvaloT z1 da z2 (13.32) Tanafaardobebidan. gvaqvs:

z1 = z – z1-/2 3n = z – z0.975 3n = 0.4 – 1.96971/2 0.201,

z2 = z + z1-/2 3n = z + z0.975 3n = 0.4 + 1.96971/2 0.599.

amitom (13.47)-dan miviRebT, rom1

11

1

2

2

z

z

e

e 0.198 da

1

12

2

2

2

z

z

e

e 0.536.

Sesabamisad, 95%-iani ndobis intervali WeSmariti -saTvis aris[0.198, 0.536].

13.7.2. oramokrefiani kriteriumebi korelaciis

koeficientisaTvis.

fiSeris gardaqmna SeiZleba ganzogadebul iqnas oramokrefiani

amocanebisaTvis. kerZod, am punqtSi Cven gvainteresebs korelaciis ko-

eficientebis tolobis sakiTxi SerCevaTa ori wyvilisaTvis. sailust-

raciod ganvixiloT Semdegi

magaliTi 13.7. ainteresebdaT aris Tu ara genetikuri kavSiri

sisxlis wnevis sidideebs Soris bavSvebsa da maT dedebs Soris. amisa-

Tvis akvirdebodnen sisxlis wnevebis sidideebs dedebsa da bavSvebSi

bavSvebis ori sxvadasxva kategoriisaTvis: maTTvis vinc namdvil deda-sTan izrdeba da maTTvis, vinc izrdeba dedinacvalTan. aRmoCnda, rom

pirveli kategoriis 1000 monacemisaTvis korelaciis koeficientis

mniSvneloba iyo 0.35, xolo meore kategoriis 100 monacemisaTvis ki0.06. aris Tu am monacemebis mixedviT genetikuri kavSiri namdvil de-

dasa da bavSvs Soris?

imisaTvis, rom pasuxi gaeces am da msgavs SekiTxvebs, davsvaT

amocana zogad statistikur formaSi. maSasadame, Cveni amocanaa nulov-

ani da alternatiuli hipoTezis garCevis sakiTxi, sadac isini ase

yalibdeba: H0 : 1 = 2, H1 : 1 2. am amocanis gadasawyvetadixilaven Sesabamisi korelaciis SerCeviTi koeficientebis z–gardaqmne-bis sxvaobas, romlisaTvisac mtkicdeba, rom H0 : 1= 2 hipoTezis

samarTlianobis SemTxvevaSi,

196

T =)3/(1)3/(1 21

21

nn

zz(13.33)

statistikas aqvs standartuli normaluri ganawileba. amitom statis-

tikuri kriteriumi ase yalibdeba:


-z1-/2 t z1-/2,

maSin mniSvnelovnebis doniT H0 hipoTezis uaryofis safuZveli aragvaqvs, winaaRmdeg SemTxvevaSi, mas uarvyofT. kriteriumis Sesabamisi


p = 2 (t), roca t 0 da p = 2 (1- (t)), roca t > 0.cxadia, igulisxmeba, rom Tavdapirveli populaciebi normalura-

daa ganawilebuli.

davubrundeT 13.7 magaliTs da vnaxoT ra pasuxs mogvcems Semo-

tanili kriteriumi. mocemulobis mixedviT, n1 = 1000, n2 = 100 da Sesa-bamisad, r1 = 0.35, r2 = 0.06. amitom

z1 = (1/2)(ln(1+ r1) - ln(1- r1)) = (1/2)(ln(1+ 0.35) - ln(1- 0.35)) 0.365,z2 = (1/2)(ln(1+ r2) - ln(1- r2)) = (1/2)(ln(1+ 0.06) - ln(1- 0.06)) 0.06

da T statistikis dakvirvebuli t mniSvneloba tolia:

t = (0.365-0.06)/(1/997+1/97)1/2 2.87.Sesabamisad, kriteriumis Sesabamisi p-mniSvneloba toli iqneba:

p = 2 (1- (t)) = p = 2 (1- (2.87)) = 0.004.maSasadame, radgan p < 0.01 davaskvniT, rom Sedegi statistikur-

ad mniSvnelovania da H0 : 1= 2 hipoTezas uarvyofT = 0.01 mniSvne-lovnebis doniT da miviRebT alternativas, rom 1 2 da e.i. genet-

ikuri kavSiri namdvili dedisa da Svilis sisxlis wnevebs Soris mar-

Tlac mWidroa.

13.8. kerZo da mravlobiTi korelacia.

zogjer saWiroa or X da Y sidides Soris kavSiris “gazomva”,romlebic damokidebulia garkveul Z1, Z2,…, Zk cvladebze. aseT SemTx-

vevebSi saWiroa korelaciis kerZo koeficientis SemoReba, romelic

Semdegnairad gansazRvreba. orive X da Y sidide ganixileba, rogorc

Z1, Z2,…, Zk damoukidebel cvladebze damokidebuli sidide da maTTvis

ganixileba wrfivi mravlobiTi regresiis modelebi. miRebul naSTiT

EX da EY wevrebs Soris korelacias uwodeben swored X da Y sidide-ebs Soris korelaciis kerZo koeficients. aseTi koeficientis cnebaCven SegviZlia SemoviRoT mravlobiTi regresiis wrfiv modelSic, ro-

desac Y sidide ganixileba, rogorc X1, X2,…, Xk damoukidebel cvlad-ebze damokidebuli sidide da Cven gvainteresebs Xi-sa da Y sidideebs

197

Soris korelaciis kerZo koeficientis gansazRvra. cxadia, rom am Sem-

TxvevaSi, EY =Y - 1 1 1( ,..., , ,..., )i i kY x x x x da EXi = Xi - 1 1 1( ,..., , ,..., )i i i kX x x x x

,

sadac 1 1 1( ,..., , ,..., )i i kY x x x x da 1 1 1( ,..., , ,..., )i i i kX x x x x

Sesabamisad aRniS-

navs Y da Xi sidideebis prognozebs X1, …, Xi-1, Xi+1,…, Xk damoukidebe-

li cvladebiT.

aseve xSirad saWiroa kavSiris dadgena Y sididesa da yvela X1,X2,…, Xk prediqtors Soris, romlebic ganixileba erT jgufad. aseT

SemTxvevaSi ganmartaven mravlobiTi korelaciis koeficients, rogorc

korelaciis koeficients Y-sa da

k

iii Xb

1

~ sidides Soris.

13.9. mravlobiTi logisturi regresia.

aqamde Cven vixilavdiT wrfiv regresiul modelebs, romlebSic

gamoZaxili warmoadgenda normalurad ganawilebul SemTxveviT sidid-

es. xSirad saWiroa msgavsi analizis Catareba gamoZaxilisaTvis, rome-

lsac aqvs binomuri ganawileba. aseT SemTxvevaSi iyeneben e.w. logist-uri regresiis modelebs, romlis mixedviTac ganawilebis p paramet-ris e.w. logisturi gardaqmnisaTvis, romelic gansazRvreba, rogorc

ln(p/(1- p)), ganixileba Semdegi wrfivi mravlobiTi regresiuli mode-

li:

ln(p/(1- p)) = a +

k

iii Xb

1

, (13.34)

sadac X1, X2,…, Xk ganixileba damoukidebel sidideebad da Y gamoZaxi-li, rogorc aRvniSneT, warmoadgns binomur SemTxveviT sidides para-

metriT p. aRvniSnoT L a +

k

iii Xb

1

. am sidides e.w. wrfiv prediqto-

rs uwodeben. maSin advilia imis danaxva, rom (13.34) eqvivalenturiaSemdegi modelisa:

p =)exp(1

)exp(

L

L

, (13.35)

saidanac cxadad Cans, rom p parametris mniSvneloba moTavsebulia

[0,1] intervalSi, miuxedavad imisa, Tu ra mniSvnelobebs Rebuloben

damoukidebeli cvladebi.

imisaTvis, rom gaviazroT regresiis bi, i = 1,2,…,k, koeficiente-bis Sinaarsi, davuSvaT, rom p parametri warmoadgens raime daavadebisSeZenis albaTobas da davuSvaT, rom ori A da B individisaTvis yvela

X1, X2,…, Xk faqtori erTnairad zemoqmedebs orive individze, garda Xi-

sa. davuSvaT, rom Xi = 1-s A individisaTvis da Xi = 0-s B individisaT-vis. maSin cxadia, rom:

198

ln(pA /(1- pA )) - ln(pB /(1- pB )) = bi, (13.36)anu

exp(bi) = (pA /(1- pA )) / (pB /(1- pB )). (13.37)

Tu gavixsenebT avadmyofobis Sansebis fardobis gamosaxulebas

SeuRlebis 22 cxrilebisaTvis 12.3 punqtidan, davaskvniT, rom

exp(bi) = ORA / ORB. (13.38)

amitom cxadia, rom RO~

= )~

exp( ib . garda amisa, ndobis interval-

is (12.19) formula Sansebis fardobis WeSmariti mniSvnelobisaTvis,SegviZlia gadavweroT logisturi regresiis terminebSi Semdegnairad:

[exp( ib~

- z1-/2 se( ib~

));exp( ib~

+ z1-/2 se( ib~

))]. (13.39)

aqedan Cven vxedavT, rom logisturi regresiis meTodebi SeiZl-

eba gamoyenebul iqnas kategoruli monacemebis analizisaTvisac.

Tu Cven romelime statistikuri paketis saSualebiT gamovTvl-

iT logisturi regresiis a~ da ib~

koeficientebs, i = 1,2,…,k, maSinSegviZlia gamovTvaloT wrfivi L prediqtoris wertilovani Sefaseba

k

iii xbaL

1

~~~ yoveli individisaTvis X1, X2,…, Xk faqtorebis mocem-

uli x1, x2,…, xk mniSvnelobebisaTvis da avagoT wrfivi prediqtorisa-

Tvis 100(1-)%-iani ndobis intervali

(L1;L2) = L~ z1-/2 se( L

~). (13.40)

cxadia, rom p parametris wertilovani Sefaseba iqneba

p~ = exp( L~

)/(1+ exp( L~

)) (13.41)

da (13.40)-dan SegviZlia avagoT 100(1-)%-iani ndobis (p1;p2) interva-li p parametris WeSmariti mniSvnelobisaTvisac. kerZod,

p1= exp(L1)/(1+ exp(L1)) da p2= exp(L2)/(1+ exp(L2)). (13.42)

amocanebi1. filtvebis funqciurobis standartul sazomad iyeneben FEV

(Forced Expiratory Volume) sidides (litrebSi). es sidide, rogorc we-

si, damokidebulia simaRlesa da asakze. imisaTvis, rom gamoekvliaT

mxolod simaRleze damokidebulebis sakiTxi, SearCies erTnairi asakis,

10-dan 15 wlamde asakis 655 biWi da miiRes cxrilSi warmodgenili

monacemebi (monacemebi gasaSualebulia yoveli 4sm-iT daSorebuli

jgufisaTvis):

simaRle, sm saSualo FEV, l simaRle, sm saSualo FEV, l

134 1.7 158 2.7138 1.9 162 3.0142 2.0 166 3.1146 2.1 170 3.4

199

150 2.2 174 3.8154 2.5 178 3.9

1.1. aageT gafantulobis diagrama da daadgineT, gvaqvs Tu ara modelis

wrfivobis daSvebis safuZveli;

1.2. umcires kvadratTa meTodis gamoyenebiT, aageT regresiis wrfe;

1.3. rogoria FEV-is prognozi 165 sm simaRlis biWebisaTvis?

1.4. aageT F -kriteriumi regresiis wrfivobisaTvis;

1.5. aageT t -kriteriumi regresiis wrfivobisaTvis;1.6. FEV-is variaciis ra nawilia damokidebuli simaRleze?

1.7. aageT 95% –iani ndobis intervali regresiis koeficientebisaTvis;

1.8. aageT 95% –iani ndobis intervali regresiis prognozisaTvis.

2. arsebobs Tu ara kavSiri martoobis grZnobasa da depresias

Soris? cxrilSi mocemulia 10 piris qulebi martoobisa da depresiis

skalebze.

Mmartoobis qula 4 27 18 7 30 12 18 23 19 12

D depresiis qula 16 37 33 23 34 32 24 29 26 26

2.1. aageT gabnevis diagrama;

2.2. gamoTvaleT korelaciis koeficenti;2.3. rogoria kavSiri am or cvlads Soris?

3. qvemoT mocemulia 10 pirisgan Semdgari SerCevis sxeulis

wonisa da TviTpativiscemis qulebi

wona ( funtebSi) 100 111 117 124 136 139 143 151 155 164

TviTpativiscema 39 47 54 23 35 30 48 20 28 46

3.1. aageT gabnevis diagrama;

3.2. arsebobs Tu ara kavSiri am monacemebs Soris?

4. axseniT ratom aris jvaredina namravlebi monacemTa or simr-

avles Soris korelaciis sazomi.

5. gansazRvreT is sami SesaZlo mizezSedegobrivi kavSiri, riT-ac SeiZleba korelaciis axsna.

6. dakavSirebulia Tu ara iumoris grZnoba fizikur janmrTel-

obasTan?

iumoris kiTxvari 36 68 61 47 35 48 42 30 56 39 51 54 60 29 65 47

fizikuri janmrTeloba 9 3 12 12 15 17 10 17 11 17 14 8 3 13 7 8

gamoiyeneT mniSnelovnobis 0.05 done.

7. cxrilSi mocemulia 15 mamakacisagan mopovebuli gabrazebis

donis qulebi da sisxlis sistoluri wnevis maCveneblebi.

brazis done – 64 30 37 39 17 28 36 34 44 32 37 47 50 41 46

sisxl. wneva– 170 132 129 144 145 122 124 145 137 115 127 148 148 144 133

SerCevis safuZvelze ra daskvnis gakeTeba SeiZleba gabrazebis donesa

da sisxlis wnevas Soris kavSiris Sesaxeb.

8. cxrilSi mocemulia abiturentebis mier zogadi unarebis

verbaluri 15 60 89 22 32 50 75 75

200

maTematika 20 58 80 12 28 46 60 89ra daskvnis gakeTeba SeiZleba verbalur da maTematikur nawilebSi mi-

Rebuli qulebs Soris kavSiris Sesaxeb.

9. fsiqologs ainteresebda arsebobs Tu ara kavSiri material-

ur uzrunvelyofasa da gamocdis win SfoTvas Soris. gamokiTxuli

iyo 10 studenti:

SfoTvis qulebi 2 5 6 10 10 17 30 10 8

materialuri uzr.

qulebi 5 20 22 30 20 55 40 30 27

SeamowmeT arsebobs Tu ara kavSiri am or sidides Soris.

10. daalageT 5 korelaciis koeficienti iseTnairad, rom pirve-li aRniSnavdes umcires wrfiv kavSirs, ukanaskneli ki udides wrfiv

kavSirs cvladebs Soris.

11. tbis ekologiuri mdgomareobis Sesaswavlad aRebuli iyo 8

sinji da maTSi gazomili iyo Jangbadis Semadgenloba. paralelurad

iTvlidnen daWerili kalmaxebis raodenobas.

Jangbadis Semadgenloba 27 12 25 26 26 16 15 23 16

kalmaxebis raodenoba 42 17 35 43 27 20 39 27 20

arsebobs Tu ara kavSiri Jangbadis Semadgenlobasa da daWerili kalma-

xebis raodenobas Soris.

12. eqims ainteresebda aris Tu ara dakavSirebuli epilefsiiTdaavadebuli bavSvebis done dRe-RameSi Mali-is epizodebTan. ori kvi-ris dakvirvebis Sedegebi moyvanilia cxrilSi:

Petit mal Episode 1 4 3 4 3 2 2 1

dopaminis done 6.5 8.3 9.7 8.1 7.5 7.0 6.1 6.3

13. zRvaSi mZime liTonebis done gansazRvravs wylis xarisxs.

baltiis zRvaze aRebuli iyo 8 sinji. X-iT aRniSnulia stronciumis

wona, xolo Y-iT ki sinjis wona.

X 4.0 5.5 4.5 4.25 6.0 5.75 4.5 4.0

Y 14 17 19 14 18 22 16 16

a) aageT gabnevis diagrama;b) gamoTvaleT korelaciis koeficienti;

g) aageT saprognozo wrfe.

14. sadazRvevo kompania ikvlevs ramdenad Zlieria kavSiri samu-

Sao dRis xangrZlivobasa da ubedur SemTxvevaTa sixSires Soris.

saaTebis raodenoba 40 32 36 44 41

SemTxvevaTa sixSire 1 1 3 8 5

risi toli iqneba ubedur SemTxvevaTa sixSire, Tu kviraSi 42 samuSao

saaTia?

15. firmis direqcias ainteresebs gaigos aris Tu ara kavSiri

misi TanamSromlebis asaksa da maT mier wlis ganmavlobaSi gacdenildReebs Soris.

201

asaki 18 26 39 48 53 58gacdenili dReebi 16 12 9 5 6 2

15.1. ipoveT korelaciis koeficienti da misi saSualebiT daadgineT

rogor kavSirSi imyofebian es cvladebi;

15.2. risi toli iqneba gacdenili dReebis raodenoba, Tu asaki iqneba

30-is toli?

l e q c i a 15.

Tavi 14. daskvniTi statistikis araparametrulimeTodebi.

14.1. Sesavali.

daskvniTi statistikis mTeli wina masala, exeboda statistikis

parametrul amocanebs, anu iseT amocanebs, romlebSic cnobili iyo

modelis ganawilebis saxe (is rogorc wesi, iyo normaluri an binomu-

ri) da statistikuri daskvnebi keTdeboda am ganawilebis ucnobi para-

metrebis Sesaxeb. Tu es daSvebebi ar keTdeba modelis Sesaxeb, an cen-

traluri zRvariTi Teoremis pirobebi darRveulia, maSin amocanebis

amosaxsnelad iyeneben statistikis araparametrul meTodebs, romelTamixedviTac modelis Sesaxeb keTdeba sakmaod susti daSvebebi da amden-

ad, hipoTezebi mowmdeba ganawilebis funqciaTa farTo klasisaTvis. ma-

galiTad, aqac SeiZleba gvainteresebdes ori populaciis saSualoebis

Sedarebis amocana. radgan aRar keTdeba populaciaTa normalurobis

daSveba, cxadia, rom populaciaTa Sesabamisi ganawilebebi SeiZleba

iyos toli saSualoebis mqone nebismieri ganawilebebi. garda amisa, am

amocanisaTvis wyvilTa t-kriteriumi optimaluri iyo mxolod da mxo-

lod im daSvebisas, roca populaciebi normaluradaa ganawilebuli.

Tu ganawilebaTa normaluroba ara gvaqvs da gadaxra am daSvebisagan

didia, maSin optimaluri aRmoCndeba am nawilSi Sesaswavli e.w. rango-brivi kriteriumebi. aq Cven SeviswavliT ramdenime aseT kriteriums.

esenia: niSnebis kriteriumi, uilkoksonis niSniani rangebis kriteriu-mi, uilkoksonis rangTa jamebis kriteriumi, kraskal-uolisis krite-riumi da spirmenis kriteriumi, romelic eyrdnoba rangobrivi korel-aciis koeficients.

14.2. niSnebis kriteriumi.

14.2.1. normaluri aproqsimacia.

magaliTi 14.1. undodaT SeedarebinaT ori A da B tipis mazisefeqturoba mziT damwvrobis sawinaaRmdegod. amisaTvis orive tipis

mazi wausves 45 adamians SemTxveviT amorCeul mklavze (erTi, erT

202

mklavze, xolo meore ki meoreze). aRmoCnda, rom 22 SemTxvevaSi B ti-pis mazwasmuli mklavi ufro wiTeli iyo, vidre A tipisa; 18 SemTxve-

vaSi piriqiT da mxolod 5 SemTxvevaSi A da B tipis mazebs hqondaTerTnairi efeqti. SeiZleba Tu ara am monacemebis mixedviT davaskvnaT,

rom damwvrobis sawinaaRmdegod A tipis mazi jobia B tipisas?aRvniSnoT, i-uri individisaTvis damwvrobis xarisxi A tipis ma-

zis xmarebisas xi-iT da igive sidide B tipis mazis xmarebisas iyos yi.

erTi SexedviT, saqme gvaqvs dawyvilebul monacemebTan, magram rogorc

vxedavT, CvenTvis cnobili araa arc xi da arc yi sidideTa mniSvnelob-

ebi TavisTavad da maSasadame, arc maTi sxvaobebi, di = xi - yi, romelic

gamoiyeneboda wyvilTa t-kriteriumis asagebad dawyvilebul monacemeb-Si. erTaderTi, rac CvenTvisaa cnobili, is aris, Tu ramden SemTxveva-

Si iyo es sidide uaryofiTi di < 0, ramdenjer – dadebiTi di > 0 da ra-mdenjer 0-is toli, anu di = 0. SemoviRoT Sesabamisi SemTxveviTi sidi-deebi Xi, Yi da Di = Xi - Yi da SevecadoT maT terminebSi gamovTqvaT is,

risi Semowmebac gvinda. bunebrivia, davuSvaT, rom D1, D2,…, Dn damouk-

idebeli da erTnairad ganawilebuli SemTxveviTi sidideebia, Tumca

Cven araferi viciT maTi FD ganawilebis saxis Sesaxeb. ganawilebis sa-

xe Cven arc gvWirdeba, radgan cxadia, rom Cveni amocana mdgomareobs

mxolod imis dadgenaSi, aqvs Tu ara efeqti romelime mazs, anu mniSv-

nelovania Tu ara gansxvaveba damwvrobaTa sawinaaRmdego efeqtebis ra-odenobebs (22–sa da 18-s) Soris, A da B tipis mazebis xmarebisas.cxadia, rom rac ufro axlos aRmoCndeba es raodenobebi (an Sesabamisi

fardobiTi sixSireebi) erTmaneTTan, miT ufro ver gavarCevdiT gansxv-

avebas mazebs Soris. magram radgan fardobiTi sixSireebi axlosaa Ses-

abamis xdomilobaTa albaTobebTan, amitom es iqneboda Sedegi {D < 0}da {D > 0} xdomilobaTa albaTobebis tolobisa. amdenad, gasagebia,

rom Cveni amocana mdgomareobs Semdegi nulovani hipoTezis Semowmeba-

Si: H0 : mD = 0, sadac mD aRniSnavs FD ganawilebis medianas, anu iseTwertils, romlisaTvisac

FD (mD) = P{D < mD } = P{D > mD}= 1 - FD (mD) FD (mD) = 1/2. (14.1)

cxadia, rom Cven saqme gvaqvs ormxriv alternativasTan: H1 : mD 0.kriteriumis asagebad SemoviRoT SemTxveviTi sidide

n

iiDIT

1

}0{ , (14.2)

sadac n aRniSnavs SerCevis moculobas (0-sagan gansxvavebul Di-ebis

raodenobas) da }0{ iDI aris SemTxveviTi sidide, romelsac {Di > 0}

xdomilobis indikators uwodeben da romelic ase ganimarteba:

0,0

0,1}0{

i

ii D

DDI

Tu

Tu(14.3)

203

maSin gasagebia, rom T gviCvenebs SerCevis im Di-ebis raodenobas,romlebic dadebiTia. amitom T SemTxveviT sidides H0 : mD = 0 hipoTez-is samarTlianobis dros aqvs binomuri ganawileba cdaTa ricxviT nda ganawilebis parametriT p = P{Di > 0}= 1/2. cxadia, rom Tu np(1- p)= n / 4 5, anu n 20, binomuri ganawilebisaTvis dasaSvebia normaluri

aproqsimacia, anu

(T – ET ) / DT = (T – n/2 ) / 4/n ~ N(0;1) (14.4)

da amitom statistikuri kriteriumi ase yalibdeba:


t > n/2 + 1/2 + z1-/2 4/n an t > n/2 - 1/2 - z1-/2 4/n , (14.5)

maSin mniSvnelovnebis doniT H0 : mD = 0 hipoTezas uarvyofT, winaa-Rmdeg SemTxvevaSi, amis safuZveli ara gvaqvs. cxadia, rom kriteriumis

Sesabamisi p-mniSvneloba tolia

p = 2(1-(( t - n/2 – 1/2) / 4/n ))), roca t n/2 da

p = 2(( t - n/2 + 1/2) / 4/n )), roca t < n/2. (14.6)

davubrundeT 15.1 magaliTs da vnaxoT, ras mogvcems kritertiu-

mi. am SemTxvevaSi, n = 45 -5 = 40 > 20, t = 18 < n/2 = 20. davuSvaT, =

0.05. maSin z0.975 = 1.96 da t1= n/2 - 1/2 - z1-/2 4/n = 20 –1/2 –1.96(40/4)1/2 13.3 da t2= n/2 + 1/2 + z1-/2 4/n = 20 +1/2 + 1.96(40/4)1/2 26.7.

vinaidan 13.3 t = 18 26.7 davaskvniT, rom = 0.05 mniSvnelovn-

ebis doniT H0 : mD = 0 hipoTezis uaryofis safuZveli ara gvaqvs. maSa-

sadame, A da B tipis mazebs aqvs erTnairi sicxis sawinaaRmdego efeq-ti. cxadia, igive pasuxs mogvcemda p-mniSvnelobis gamoTvlac. marT-lac, radgan t = 18 < n/2 = 20, (14.6)-is meore tolobidan p-mniSvneloba

tolia p = 2(( t - n/2 + 1/2) / 4/n )) = 2((20 - 18 + 1/2) /(40/4)1/2)) 0.635>0.05.

14.2.2. zusti meTodi.

zusti meTodi gulisxmobs p-mniSvnelobis gamoTvlas zustad

da is upiratesad gamoiyeneba im SemTxvevaSi, roca dauSvebelia binomu-ri ganawilebis normaluriT aproqsimacia, anu roca n < 20. am SemTxve-vaSi, p-mniSvneloba gamoiTvleba formuliT:

p = 2P{bin(n,1/2) t} =

tk

nknC )2/1( , roca t > n/2 da

p = 2P{bin(n,1/2) t} =

tk

nknC )2/1( , roca t < n/2. (14.7)

204

14.3. uilkoksonis niSniani rangebis kriteriumi.

magaliTi 14.2. davuSvaT, rom magaliT 14.2-Si A da B tipis ma-zebis Sedareba xdeba damwvrobis xarisxebis mixedviT, romelic izom-

eba 10 baliani skaliT da monacemebi warmodgenilia Semdegi cxrilis

saxiT:

sxvaoba, di sixSire, fi sxvaoba, dI sixSire, fI

-10 0 10 0-9 0 9 0-8 1 8 0-7 3 7 0-6 2 6 0-5 2 5 0-4 1 4 0-3 5 3 2-2 4 2 6-1 4 1 10 22 180 5am cxrilSi mocemuli sidideebis gasaazreblad ganvixiloT mag-

aliTad, di = -6 da Sesabamisad, fi = 2 SemTxveva. es imas niSnavs, rom oradamians A tipis mazis xmarebisas cal xelze da B tipisas – meoreze,

aRmoaCnda damwvrobis xarisxebs Soris sxvaoba 6-is toli, Tanac Atipis mazis sasargeblod. Cveni amocana kvlav mdgomareobs am monaceme-

bze dayrdnobiT gavarkvioT mazebis efeqturobis tolobis sakiTxi.

cxadia, rom hipoTeza da alternativa aq isev ise yalibdeba:

H0 : mD = 0 da H1 : mD 0.erTi SexedviT kvlav SeiZleba mogveCvenos, rom aq SesaZlebelia

wyvilTa t-kriteriumis gamoyeneba, magram SevniSnoT, rom monacemebi ga-zomilia e.w. ordinalur skalaSi, anu isini dalagebulia, magram ar

gaaCniaT konkretuli ricxviTi mniSvneloba. sxva sityvebiT rom vTqv-

aT, di = -6 sulac ar niSnavs imas, rom 6-baliani sxvaoba 6-jer metia1-balian di = -1 sxvaobaze, es dalageba gulisxmobs mxolod, rom Atipis mazis sasargeblod -6 jobia -5-s, -5 jobia -4-s da a.S. amitomaseT SemTxvevaSi gamoiyeneba uilkoksonis niSniani rangebis kriteriu-mi, romelic Semdegnairad igeba: pirvel rigSi, gamovricxoT is monace-

mebi, sadac di = 0; Semdeg SevxedoT erTnairi modulebis mqone di-ebis

jamur sixSireebs, magaliTad, im monacemebis raodenoba, sadac | di | = 1,aris 4 + 10 = 14; im monacemebis raodenoba, sadac | di | = 2, aris 4 + 6 =10 da a.S. im monacemebis jgufs, sadac | di | = 1, da monacemebis raoden-oba aris 14, Seesabameba rangebis diapazoni 1-dan 14-mde, amitom amjgufs mivaniWoT saSualo rangi (1 + 14)/2 = 7.5, analogiurad, im jgu-

205

fisaTvis, sadac | di | = 2 da monacemebis raodenoba aris 10, Seesabamebarangebis diapazoni 14+1=15-dan 14+10 = 24-mde, amitom am jgufs mivan-iWoT saSualo rangi (15 + 24)/2 = 19.5 da a.S. aseTi wesiT miRebuli

ricxvebi CavweroT cxrilis saxiT. miviRebT:

| dI | sixSire rangis diapazoni saSualo rangi

10 0 + 0 = 0 – –9 0 + 0 = 0 – –8 1 + 0 = 1 40 40.07 3 + 0 = 3 37 – 39 38.06 2 + 0 = 2 35 – 36 35.55 2 + 0 = 2 33 – 34 33.54 1 + 0 = 1 32 32.03 5 + 2 = 7 25 – 31 28.02 4 + 6 = 10 15 – 24 19.51 4 + 10 = 14 1 – 14 7.5kriteriumis statistikad iReben statistika R+-s, romelic wa-

rmoadgens dadebiTniSniani di-ebis Sesabamisi rangebis jams, romlis

ricxviTi maxasiaTeblebic tolia:

ER+ = n(n + 1)/4 da DR+ = n(n + 1)(2n + 1)/24 (14.8)

da romlisaTvisac mtkicdeba, rom Tu aranulovani di-ebis raodenoba 16, maSin R+-is ganawileba aproqsimirdeba normaluri ganawilebiT, anu

T =24/)12)(1(

2/14/)1(2/1

nnn

nnR

DR

ERR ~ N(0;1). (14.9)

SevniSnoT, rom es Tanafardoba samarTliania maSin, rodesac ara

gvaqvs erTnairi modulebis mqone sxvaobaTa jgufebi. im SemTxvevaSi,

roca es piroba darRveulia (rogorc es Cvens magaliTSia) (14.9)-Si

dispersiis gamosaxuleba unda Seswordes. kerZod, aseT dros

DR+ = n(n + 1)(2n + 1)/24 -

g

iii tt

1

3 2/)( , (14.10)

sadac ti -iT aRniSnulia i-ur jgufSi sxvaobaTa raodenoba, romlebsac

erTnairi modulebi aqvT da g aRniSnavs aseTi jgufebis saerTo raod-enobas.

statistikuri kriteriumi ase yalibdeba:


| t | > z1-/2, (14.11)

maSin mniSvnelovnebis doniT H0 : mD = 0 hipoTezas uarvyofT, winaa-Rmdeg SemTxvevaSi, amis safuZveli ara gvaqvs. cxadia, rom kriteriumis

Sesabamisi p-mniSvneloba tolia

p = 2(1 - ( t)). (14.12)

206

davubrundeT 15.2 magaliTs. am SemTxvevaSic, n = 45 - 5 = 40 16da amitom dasaSvebia normaluri aproqsimacia. ukanaskneli cxrilidan

gamovTvaloT R+-is dakvirvebuli mniSvneloba. gvaqvs:

r+ = 107.5 + 619.5 +228.0 = 248;(14.10) formulis mixedviT gamovTvaloT Sesworebuli dispersiis mni-Svneloba:

DR+ = n(n + 1)(2n + 1)/24 -

g

iii tt

1

3 2/)( = 404181/24 – [(143-14) +

+ (103-10) + (73-7) + (23- 2) + (23- 2) +(33- 3)]/2 = 3489.amitom sabolood, (14.9) formulis mixedviT gamoTvlili T

statistikis dakvirvebuli t mniSvneloba tolia:

t =

DR

nnr 2/14/)1(= (|248 - 4041/4|-1/2)/34891/2 = 2.73.

(14.12)-dan vpoulobT kriteriumis Sesabamis p-mniSvnelobas:

p = 2(1-( t)) = 2(1-(2.73)) = 0.006,rac cxadia, metyvelebs Sedegis statistikur mniSvnelovnebaze da maS-

asadame, H0 : mD = 0 hipoTezas uarvyofT da davaskvniT, rom A tipismazi jobia B tipisas. rogorc vxedavT, 14.1 magaliTisagan gansxvaveb-

iT, miviReT sawinaaRmdego daskvna. es metyvelebs imaze, rom rac ufro

meti informaciaa xelmisawvdomi, daskvna miT ufro zusti iqneba: 14.1magaliTSi mocemuli informaciiT laparaki iyo mxolod imaze, Tu da-

mwvrobis xarisxebs Soris sxvaoba dadebiTia Tu uaryofiTi da amitom-

ac iq miRebulze ukeTesi daskvnis miReba SeuZlebelia, maSin roca

14.2 magaliTSi damwvrobis xarisxebi xasiaTdeba kidev maTi siaxloviT;

amitomac informaciis damatebam 14.2 magaliTSi Tvisobrivad Secvala

statistikuri daskvna.

SevniSnoT, rom am kriteriumis statistikad Cven gamoviyeneT

statistika R+-s, romelic warmoadgens dadebiTniSniani di-ebis Sesaba-

misi rangebis jams. bunebrivia ismis kiTxva: ra moxdeboda, rom agveRo

R -, romelic uaryofiTniSniani di-ebis Sesabamisi rangebis jamia? amis

dasadgenad unda aRvniSnoT, rom

R + + R - = n(n + 1)/2, (14.13)

saidanac cxadia, rom | R - - n(n + 1)/4 | = | R + - n(n + 1)/4 | da amitom

(14.9) statistikis mniSvneloba ar icvleba.

14.4. uilkoksonis rangTa jamis kriteriumi.

wina punqtSi Seswavlili niSniani rangebis kriteriumi, garkveu-

li azriT, aris t-kriteriumis araparametruli analogi dawyvilebuli

monacemebisaTvis. axla Cven gvinda SeviswavloT imave t-kriteriumis

207

araparametruli analogi damoukidebeli SerCevebisaTvis. saxelad amkriteriums hqvia uilkoksonis rangTa jamis kriteriumi.

magaliTi 14.3. iTvleba, rom sxvadasxva genotips aqvs Tvalis

baduris anTebis (RP – Retinitis Pigmentosa) ganviTarebis sxvadasxva sis-wrafe: dominanturi forma, romlis drosac daavadeba nela viTardeba;

recesiuli forma, romlis drosac daavadeba ufro nela viTardeba da

SL (Sex-linked) forma, romlis drosac daavadeba viTardeba swrafad.

am hipoTezis Sesamowmeblad daakvirdnen 10-19 asakobrivi jgufis moz-

rdebSi VA (vizual acuity) sidideebs, romlebsac hqondaT RP-is sxvadas-xva genotipi da miiRes cxrilSi warmodgenili monacemebi:

VA dominanturi forma SL forma

20 – 20 5 120 – 25 9 520 – 30 6 420 – 40 3 420 – 50 2 820 – 60 0 520 – 70 0 220 – 80 0 1

= 25 = 30rogor SeiZleba es monacemebi gamoviyenoT imis Sesamowmeblad, rom

VA sidideebis ganawilebaTa medianebi ori formisaTvis sxvadasxvaa?

aRvniSnoT mD–Ti da mSL–iT Sesabamisad pirveli da meore amok-refis Sesabamisi populaciebis medianebi. maSin nulovani da alternat-

iuli hipoTeza ase yalibdeba: H0 : mD = mSL da H0 : mD mSL. wina punq-

tSi ganxiluli amocanis msgavsad, aqac ar SeiZleba t–kriteriumis ga-moyeneba, imis gamo, rom VA sidideebis ricxviTi mniSvnelobebi Cven ar

viciT. aseT SemTxvevebSi gardauvalia araparametruli meTodis gamoye-

neba. uilkoksonis rangTa jamis kriteriumi, romlis gamoyenebasac aq

vapirebT, warmoadgens swored aseT meTods. is damyarebulia monacemTarangebze da Semdegnairad igeba: gavaerTianoT orive amokrefa da mivaw-

eroT rangebi VA sidideebis zrdadobis mixedviT (saukeTeso VA sidi-

didan (20-20), Zalze cudamde (20-80)). magaliTad, gaerTianebuli Se-

rCevisaTvis pirveli striqonidan miviRebT, rom monacemebis raodenobaa

5 + 1 = 6, amitom mas miewereba rangobrivi 1 – 6 intervali da saSualo

rangi iqneba (1 + 6) / 2 = 3.5, meore striqonisaTvis Sesabamisad gveqneba9 + 5 = 14, rangebis intervalia 7 – 20 (wina intervalis udidess + 1,anu 6 + 1 = 7 da wina intervalis udidess + 14, anu 6 + 14 = 20) da a.S.

sabolood, miRebuli Sedegebi warmovadginoT Semdegi cxrilis

saSualebiT:gaerTianebuli SerCeva rangebis diapazoni saSualo rangi

5 + 1 = 6 1 – 6 3.5

208

9 + 5 = 14 7 – 20 13.56 + 4 = 10 21 – 30 25.53 + 4 = 7 31 – 37 34.02 + 8 = 10 38 – 47 42.50 + 5 = 5 48 – 52 50.00 + 2 = 2 53 – 54 53.50 + 1 = 1 55 55.0 = 55

kriteriumis statistikad iReben statistika R1-s, romelic wa-rmoadgens I amokrefis Sesabamisi rangebis jams, romlis ricxviTi max-

asiaTeblebic tolia:

ER1 = n1 (n1 + n2 + 1) / 2 da DR1 = n1 n2 (n1 + n2 + 1) / 12 (14.14)

da romlisaTvisac mtkicdeba, rom Tu umciresi SerCevis moculoba,

anu min(n1, n2) 10 da SerCevaTa Sesabamisi populaciebi uwyvetadaa ga-nawilebuli, maSin R1-is ganawileba aproqsimirdeba normaluri ganawi-

lebiT, anu

T =12/)1(

2/12/)1(2/1

2121

2111

1

11

nnnn

nnnR

DR

ERR ~ N(0;1). (14.15)

SevniSnoT, rom es Tanafardoba samarTliania maSin, rodesac ga-

erTianebul amokrefaSi elementi xvdeba erTxel. im SemTxvevaSi, roca

es piroba darRveulia (rogorc es Cvens magaliTSia) (14.14)-Si dispe-

rsiis gamosaxuleba unda Seswordes. kerZod, am SemTxvevaSi

DR1 = (n1 n2 / 12) [n1 + n2 + 1 -)1()(

)(

2121

1

3

nnnn

ttg

iii

], (14.16)

sadac ti-iT aRniSnulia gaerTianebuli SerCevis i-ur striqonSi mdgomiricxvi, xolo ti-Ti – striqonebis saerTo raodenoba. SevniSnoT, rom

(14.16) moicavs (14.14)-s, anu Tu (14.16)-Si yvela ti = 1, maSin (14.16)-dan miviRebT (14.14)-s.



| t | > z1-/2, (14.17)

maSin mniSvnelovnebis doniT H0 : mD = mSL hipoTezas uarvyofT, win-aaRmdeg SemTxvevaSi, amis safuZveli ara gvaqvs. cxadia, rom kriteriu-

mis Sesabamisi p-mniSvneloba tolia

p = 2(1-( t)). (14.18)

davubrundeT 15.3 magaliTs. am SemTxvevaSic, n2 = 30 > n1 = 25 10da amitom dasaSvebia normaluri aproqsimacia. ukanaskneli cxrilidan

gamovTvaloT R1-is dakvirvebuli mniSvneloba. gvaqvs:

209

r1 = 53.5 + 913.5 + 625.5 + 334.0 + 242.5 = 479;cxadia, rom ER1 = 25(25 + 30 + 1) / 2 = 700.

(14.16) formulis mixedviT gamovTvaloT dispersia:

DR1 = (n1 n2 / 12) [n1 + n2 + 1 -)1()(

)(

2121

1

3

nnnn

ttg

iii

] =

= (2530/12) [56 - {(63-6) + (143-14) + (103-10) + (73- 7) + (103- 10) ++ (53- 5) + (23- 2)}/(5554)] = 3386.74.

amitom sabolood, (14.15) formulis mixedviT gamoTvlili Tstatistikis dakvirvebuli t mniSvneloba tolia:

t =1

11 2/1

DR

ERr = (|479 - 700|-1/2)/3386.741/2 = 3.79.

(14.18)-dan vpoulobT kriteriumis Sesabamis p-mniSvnelobas:

p = 2(1-( t)) = 2(1-(3.79)) < 0.001,rac cxadia, metyvelebs Sedegis statistikur mniSvnelovnebaze da maS-

asadame, H0 : mD = mSL hipoTezas uarvyofT da davaskvniT, rom daavadeb-

is dominanturi da SL jgufebi mniSvnelovnad gansxvavdebian: dominan-

tur jgufs aqvs ukeTesi VA.

SevniSnoT, rom kriteriumiT sargeblobisas Cven gamoviyeneT pi-

roba, rom min(n1, n2) 10. ra moxdeba im SemTxvevaSi, Tu es piroba da-rRveulia? aseT SemTxvevaSi, R1 statitikis dakvirvebul r1 mniSvnelo-

bas adareben specialur cxrilebSi mocemul zeda da qveda dasaSveb

mniSvnelobebs (romlebic tabulirebulia sxvadasxva -saTvis) da Tur1-is mniSvneloba ar moxvdeba am intervalSi, maSin H0 hipoTezas uar-

yofen, winaaRmdeg SemTxvevaSi, kvlav iTvleba, rom hipoTezis uaryofis

safuZveli ara aqvT.

garda amisa, SevniSnavT, rom am kriteriums xSirad man-uitniskriteriumsac adareben, romlis Sesabamisi statistikac agebulia im

(xi, yj) wyvilebis raodenobaze, romelTaTvisac xi < yj. Cven aq ar moviyv-

anT am kriteriumis agebis proceduras, aRvniSnavT mxolod, rom es

ori kriteriumi arsebiTad eqvivalenturia im azriT, rom maTi Sesabam-

isi p-mniSvnelobebi emTxveva erTmaneTs.

bolos gavixsenoT, rom uilkoksonis rangTa jamis kriteriumis

gamoyenebis erT-erT ZiriTad winapirobas warmoadgens populaciaTa

uwyvetad ganawilebuloba. am mxriv unda aRiniSnos, rom makneilma gam-

oiyena es kriteriumi diskretuli, kerZod, dajgufebuli monacemebisa-

Tvis normaluri ganawilebidan da aCvena, rom miRebuli kriteriumissimZlavre Zalian cotaTi iklebs igive kriteriumis simZlavresTan Se-

darebiT Cveulebrivi (daujgufebeli) normaluri populaciebidan miRe-

buli SerCevebisaTvis. ase, rom am SemTxvevaSi SesaZlebelia uilkokso-

210

nis rangTa jamis kriteriumis gamoyeneba dajgufebuli monacemebisaTvi-sac.

14.5. Tanadoba uilkoksonis rangTa jamis kriteriumsa da

2 -kriteriums Soris.

wina nawilSi ganxiluli uilkoksonis rangTa jamis kriteriumi

warmoadgens trendis gamosavleni 2–kriteriumis kerZo SemTxvevas

2c cxrilebSi, romelic Cven ganvixileT 12.7 punqtis bolos. amis

dasanaxad, 14.5 magaliTSi mocemuli monacemebi warmovadginoT Semdegi

28 cxrilis saxiT:

rogorc vxedavT, VA sidideebis mixedviT monacemebi kategorizebulia

rva binomuri SerCevis saxiT, xolo saSualo rangebi miCneulia qulis

sidideebad.

sazogadod, rogorc gvaxsovs, trendis gamosavlenad 2c cxri-

ls hqonda aseTi saxe:

aRvniSnoT pi-iT i-ur SerCevaSi daavadebis albaToba da davuSv-

aT, rom pi = p + Si. maSin SerCevaTa erTgvarovnebis hipoTeza gamoiT-

qmeba ase: H0 : = 0, xolo alternativa – H1 : 0. rogorc gvaxsovs,2 kriteriumi eyrdnoboda X 2 = (|O - E| -1/2)2 / V statistikasa da imfaqts, rom H0 hipoTezis samarTlianobis SemTxvevaSi, T ~ 2 (1), sad-ac

211

O =1

c

i ii

x S , E =

1

c

i ii

xx S

n

, V = 2 2

1 1

( )( ) /

( 1)

c c

i i i ii i

x n xn S n S n

n n

(14.19)

da gadawyvetilebis miRebis wesi aseTi iyo, rom Tu X 2 statistikis

dakvirvebuli x2 mniSvneloba metia 21, 1- -ze, anu x2 > 2

1, 1-, maSin H0

hipoTezas uarvyofdiT, xolo winaaRmdeg SemTxvevaSi, amis safuZveli

ar gvqonda.

amave terminebSi, uilkoksonis kriteriumis statistika ase gam-

oiyureba:

T =

))1(/()(112

)(

2/12/)1(

1

3

1

nnnnnxnx

nxR

c

iii

(14.20)

xolo gadawyvetilebis miRebis wesi aseTia: Tu T statistikis dakvi-

rvebuli t mniSvneloba metia z1-/2 -ze, anu t > z1-/2, maSin H0 hipoTezas

uarvyofT, xolo winaaRmdeg SemTxvevaSi, amis safuZveli ara gvaqvs.

vnaxoT ras gvaZlevs 2 kriteriumi magaliT 14.5-Si. zemoT moy-

vanili 28 cxrilis mixedviT gvaqvs:

O = 53.5+913.5+625.5+334.0+242.5 = 479;E=(25/55)[63.5+1413.5+1025.5+734.0+1042.5+550.0+253.5+155.0= 700;V= (2530/5554)[63.52+1413.52+1025.52+734.02+1042.52+550.02+253.52+

+155.02- 15402 /55] = 3386.74da maSasadame, vRebulobT, rom x2 = (| 479-700 | -1/2)2 / 3386.74 = 14.36.Sesabamisi p-mniSvnelobaa p = P{2

1 > x2} = P{21 > 14.36} < 0.001 da H0

hipoTezas uarvyofT. garda imisa, rom miviReT igive Sedegi, rac uil-

koksonis kriteriumma mogvca, SevniSnoT, rom x2 = t2.

tipiurad es ase moxdeba im SemTxvevaSi, roca qulis sidided

aviRebT saSualo rangebs, magram Tu qulebad avirCevT sxva sidideebs,

maSin x2 t2, Tumca hipoTezaTa Semowmebis TvalsazrisiT, Sedegi igive

darCeba. SevniSnoT, rom X2 statistikis dispersiis (14.19) gamosaxul-ebaSi adre Cven gvqonda

D =

nSnSnn

xnx c

iii

c

iii /

)(2

11

22

,

rac didi n-ebisaTvis umniSvnelod gansxvavdeba V-sagan. es gamowveu-lia imiT, rom ukanasknel gamosaxulebaSi dispersiis gamosaTvlelad

gamoiyeneba binomuri ganawileba, xolo (14.19)-Si ki – hipergeometriu-

li ganawileba, romelic ufro esadageba aRwerili tipis amocanebs.

14.6. kraskel-uolisis kriteriumi.

zogierT situaciaSi Cven gvWirdeba orze meti populaciis saS-

ualos Sedareba, magram dispersiuli analizis (ANOVA) modelebisag-

212

an gansxvavebiT populaciaTa ganawilebebi SeiZleba aRar iyos normal-uri ganawilebebi. aseT SemTxvevebSi, dispersiuli analizis meTodebi

ar gamodgeba da populaciaTa saSualoebis Sesadareblad saWiroa pop-

ulaciaTa ganawilebebisagan Tavisufali (araparametruli) proceduris

ageba. swored aseT proceduras warmoadgens uilkoksonis kriteriumis

ganzogadeba, romelic literaturaSi kraskel-uolisis kriteriumissaxeliTaa cnobili. ZiriTad daSvebad am kriteriumis gamoyenebisas

kvlav rCeba populaciaTa uwyvetad ganawilebulobis daSveba.

magaliTi 14.4. cnobilia, rom Tvalis metabolizmze gavlenas

axdens araqidonis mJava (Arachidonic acid). kerZod, cnobilia, rom sxva

efeqtebTan erTad, is iwvevs Tvalis quTuTos daxurvas, itching da misgaTavisuflebas. oTxi sxvadasxva wamlis efeqtianobis Sesadareblad

Seiswavleboda maTi moqmedeba albinosur kurdRlebSi araqidonis mJav-

is Seyvanis Semdeg. yovel jgufSi aerTianebdnen eqvs kurdRels da Ti-

Toeul cxovels erT TvalSi awveTebdnen Sesabamis wamals, xolo meo-

reSi – uvnebel saSualebas. aTi wuTis Semdeg orive TvalSi awveTebd-

nen araqidonis mJavas. amis Semdeg axdendnen orive Tvalis kontrols

15 wuTis Semdeg quTuTos daxurvidan. yovel cdaSi aRiricxeboda or-

ive Tvalis quTuTo sam baliani, 0-dan 3-mde skaliT, sadac 0 niSnavda

– Tvali mTlianad Riaa, 3 niSnavda, rom Tvali mTlianad daxurulia,

xolo 1 da 2 ki niSnavda garkveul Sualedur mdgomareobebs. efeqtu-robis sazomad (x) iyenebdnen sxvaobas sacdel (wamlian) TvalSi Tval-

is dafarvis xarisxs gamoklebuli igive sididis mniSvneloba meore

TvalSi. x-is didi mniSvneloba miuTiTebs wamlis efeqtianobas. monace-

mebi moyvanilia Semdeg cxrilSi:

kriteriumis asagebad aerTianeben yvela jgufs da TiToeul in-

divids miaweren Sesabamis rangs, an toli monacemebis SemTxvevaSi, ran-

gebis diapazons da Semdeg iTvlian rangebis saSualo iR mniSvnelobas

imave wesiT, rogorc es gakeTebuli iyo uilkoksonis kriteriumis age-

bisas. cxrilSi es ukve gakeTebulia (SeamowmeT siswore!). kriteriumis

asagebad adareben jgufebis jamur Ri rangebs. Tu isini “Sorsaa” erTm-

aneTisagan, maSin populaciaTa erTgvarovnebis (saSualoTa tolobis)

213

H0 hipoTezas uaryofen, xolo winaaRmdeg SemTxvevaSi, amis safuZveliar arsebobs. imisaTvis, rom gavarkvioT ras niSnavs sityva “Sors”, mo-

viyvanoT kriteriumis statistikis saxe da TviTon kriteriumi:

)/()(1/)1(3)1(

12 3

1

3

1

2

nnttnn

R

nnT

g

jjj

k

i i

i , (14.21)

sadac k aRniSnavs SerCevaTa raodenobas, g – toli rangebis mqone da-

kvirvebaTa saerTo raodenobas, tj aRniSnavs monacemebis raodenobas j-ur klasterul (toli rangobrivi diapazonis mqone) jgufSi dabol-

os,

k

iinn

1

- aris gaerTianebuli SerCevis moculoba.

SevniSnoT, rom Tu yvela tj = 1, maSin (14.21) wiladis mniSvneli

1-is tolia da gamosaxuleba martivdeba (gvrCeba mxolod mricxveli)

da Rebulobs Semdeg saxes:

)1(3)1(

12

1

2

nn

R

nnT

k

i i

i . (14.22)

garda amisa, Tu SerCevaTa moculobebi tolia (rogorc es 14.4

magaliTSia), maSin ni = n/k, imave (14.21) wiladis mricxveli martivdebada vRebulobT:

)/()(1/)1(3)1(

12 3

1

3

1

22

nnttnRnn

kT

g

jjj

k

ii . (14.23)

statistikuri kriteriumi eyrdnoba im faqts, rom H0 hipoTez-

is samarTlianobisas T ~ 2(k -1) da maSasadame, is ase yalibdeba:

Tu dasaxelebuli mniSvnelovnebis donisaTvis T statistikis

dakvirvebuli t mniSvneloba metia 2k - 1, 1- -ze, maSin H0 hipoTezas uar-

vyofT, xolo winaaRmdeg SemTxvevaSi, amis safuZveli ara gvaqvs. krit-

eriumis Sesabamisi p-mniSvneloba tolia p = P{2k – 1 > t} da kriteriumi

gamoiyeneba maSin, roca yvela ni 5.davubrundeT 14.4 magaliTs da vnaxoT, ras mogvcems kriteriu-

mi. kriteriumis statistikis mniSvnelobis gamosaTvlelad moxerxebu-

lia monacemebi warmovadginoT Semdegi cxrilis saxiT:

qula sixSire rangebis diapazoni saSualo rangi

-1 1 1 1.00 5 2 – 6 4.0

+1 5 7 – 11 9.0+2 4 12 – 15 13.5+3 9 16 – 24 20.0gamovTvaloT pirvel rigSi jgufebis jamuri Ri rangebi:

R1 = 13.5 + 20.0 + 20.0 + 20.0 + 20.0 + 4.0 = 97.5;R2 = 9.0 + 20.0 + 9.0 + 13.5 + 13.5 + 20.0 = 85.0;R3 = 20.0 + 9.0 + 13.5 + 9.0 + 20.0 + 20.0 = 91.5;

214

R4 = 9.0 + 4.0 + 4.0 + 4.0 + 4.0 + 1.0 = 26.0.(14.23) gamosaxulebis mricxveli tolia:

((124)/(24225))[97.52 + 852 + 91.52 + 26.02] - 325 = 10.932,xolo mniSvneli Cvens mier Sedgenili cxrilis mixedviT tolia:

)/()(1 3

1

3 nnttg

jjj

= 1-[(53-5) + (53-5) + (43-4) + (93-9)]/ (243-24) = 0.926.

amitom sabolood, t = 10.932/0.926 = 11.804. radgan k = 4, 2(3)ganawilebis cxrilebidan = 0.01-saTvis gvaqvs 2

k-1, 1- = 23, 0.99 = 11.34

da radgan 23, 0.99 = 11.34 < 11.804 < t, = 0.01-saTvis H0 hipoTezas uar-

vyofT da davaskvniT, rom wamlebs aqvT anTebis sawinaaRmdego sxvadas-

xva efeqti.

imis dasadgenad, Tu romeli populaciebis saSualoebi gansxvav-

deba, anu aris Tu ara m-uri da l-uri jgufebis saSualoebi toli,

iyeneben Semdeg statistikas

lmlmlm nn

nnRRz

11

12

)1(/)(, (14.24)

da gadawyvetilebis miRebis Semdeg wess:

Tu | zm,l | > *1 z , maSin m-uri da l-uri jgufebis saSualoebis

tolobis hipoTezas uaryofen, winaaRmdeg SemTxvevaSi, amis safuZveli

ar arsebobs. aq * = /(k(k-1)) da jjj nRR / .

14.4 magaliTisaTvis, * = /(4(4-1)) = /12 = 0.0042, roca =0.05. radgan yvela nj = 6, gvaqvs

1R = R1/6 = 97.5/6 = 16.25; 2R = R2/6 = 85/6 = 14.17;

3R = R3/6 = 91.5/6 = 15.25; 4R = R4/6 = 26/6 = 4.33.

amitom

z1,2 = (16.25 – 14.17)/(2425/12)(1/6+1/6) = 0.51;z1,3 = (16.25 – 15.25)/(2425/12)(1/6+1/6) = 0.24;z1,4 = (16.25 – 4.33)/(2425/12)(1/6+1/6) = 2.92;z2,3 = (14.17 – 15.25)/(2425/12)(1/6+1/6) = -0.27;z2,4 = (14.17 – 4.33)/(2425/12)(1/6+1/6) = 2.41;z3,4 = (15.25 – 4.33)/(2425/12)(1/6+1/6) = 2.67.

normaluri ganawilebis cxrilebidan vpoulobT, romz1- 0.0042 = z 0.9958 = 2.635

da radgan am kritikul mniSvnelobaze absoluturi sididiT mxolod

z1,4 da z3,4 mniSvnelobebia meti, davaskvniT, rom I da III tipis wamlebs

IV tipis wamalTan SedarebiT gaaCniaT mniSvnelovani efeqtebi, xolodanarCeni Sedegebi statistikurad umniSvneloa.

215

14.7. spirmenis rangobrivi korelaciis koeficienti.

zogjer saWiroa SerCevebze dayrdnobiT or populacias Soris

kavSiris xarisxis gansazRvra, magram es populaciebi araa normalurad

ganawilebuli an gazomvebi Catarebulia ordinalur skalaze da amit-

om ar gamodgeba korelaciuri analizis ukve Seswavlili meTodebi. am

SemTxvevaSi iyeneben spirmenis rangobrivi korelaciis koeficients.magaliTi 14.5. 1952 wlidan aSS-Si axalSobilis fizikuri kon-

diciis gasazomad dabadebidan 1 da 5 wuTis Semdeg farTod gamoiyenebaafgaris (Afgar) qulaTa meTodi, romelic dgindeba xuTi komponentis

SekrebiT, romlebic sam balian skalebSi (0-dan 2-mde) izomeba. es meT-

odi warmodgeba aseTi cxrilis saxiT:

davuSvaT, 24 gvaqvs axalSobilis afgaris qulebze dakvirvebebis

Sedegebi dabadebidan 1 da 5 wuTis Semdeg. Cveni amocanaa monacemebzedakvirvebebis safuZvelze davadginoT am qulebs Soris kavSiris mniSv-

nelovneba. rogor SeiZleba es gakeTdes? monacemebi warmodgenilia Sem-

degi cxrilis saxiT:axalSobili afgaris qula (1 wT) afgaris qula (5 wT)

1 10 102 3 63 8 94 9 105 8 96 9 107 8 98 8 99 8 910 8 911 7 912 8 913 6 914 8 1015 9 1016 9 10

216

17 9 1018 9 919 8 1020 9 921 3 322 9 923 7 1024 10 10

spirmenis rangobrivi korelaciis koeficienti ganimarteba Cveu-

lebrivi korelaciis koeficientis msgavsad, mxolod im gansxvavebiT,

rom L sidideebi iTvleba monacemTa rangebisaTvis da ara TviT monace-

mebisaTvis:

yyxx

xyS

LL

Lr

. (14.25)

am koeficientis ganxilvis azrianoba dafuZnebulia im faqtze,

rom Tu X da Y sididebs Soris korelacia srulia, anu r = 1, maSin rS

= 1 da roca r < 1 Sesabamisad, rS < 1. rS -is gamosaTvlelad monacemebis

cxrils mivaweroT Sesabamisi rangebis svetebi:

axalS-

obili

qula (1 wT) qula (5 wT) rangi (1 wT) rangi (5 wT)

1 10 10 23.5 19.52 3 6 1.5 2.03 8 9 10.0 8.54 9 10 18.5 19.55 8 9 10.0 8.56 9 10 18.5 19.57 8 9 10.0 8.58 8 9 10.0 8.59 8 9 10.0 8.510 8 9 10.0 8.511 7 9 4.5 8.512 8 9 10.0 8..513 6 9 3.0 8.514 8 10 10.0 19.515 9 10 18.5 19.516 9 10 18.5 19.517 9 10 18.5 19.518 9 9 18.5 8.519 8 10 10.0 19.520 9 9 18.5 8.521 3 3 1.5 1.022 9 9 18.5 8.523 7 10 4.5 19.524 10 10 23.5 19.5

217

124R =1000/24 2

24R =300/24

am sidideebs Soris korelaciis koeficientis gamoTvla gvaZl-

evs rS = 0.593. amitom rac Cven gvainteresebs aris is, Tu ramdenad mni-Svnelovania es Sedegi. mtkicdeba, rom

21

2

S

SS

r

nrT

(14.26)

statistikas aqvs t(n-2)-ganawileba da amitom kavSiris mniSvnelovnebis


Tu dasaxelebuli mniSvnelovnebis donisaTvis TS statistikis

dakvirvebuli tS mniSvneloba akmayofilebs pirobas

-tn-2, 1-/2 < tS < tn-2, 1-/2,

maSin mniSvnelovnebis doniT H0 hipoTezis (niSanTa damoukideblob-

is hipoTezis) uaryofis safuZveli ar gvaqvs, xolo winaaRmdeg SemTx-

vevaSi, mas uarvyofT. kriteriumis Sesabamisi p-mniSvneloba gamoiTvle-

ba Semdegi wesiT:

p = 2P{ tn-2 < tS }, Tu tS < 0 da p = 2P{ tn-2 > tS }, Tu tS > 0.kriteriumis gamoyeneba SeiZleba mxolod maSin, roca n 10.davubrundeT 14.5 magaliTs da vnaxoT ras mogvcems kriteriumi.

am SemTxvevaSi, n = 24, rS = 0.593, amitom statistikis dakvirvebuli

mniSvneloba tolia:

tS = (0.593221/2)/(1-0.5932)1/2 = 3.45.t(22)-ganawilebis cxrilebidan gvaqvs: t22, 0.995 = 2.819 da t22, 0.9995 =

3.792, saidanac davaskvniT, rom0.001 = 2(1-0.9995) < p < 2(1-0.995) = 0.01

da maSasadame, dakvirvebadi qulebis rangebs Soris aris mniSvnelovanikorelacia.

SevniSnoT, rom Tu darRveulia piroba iyeneben specialur

cxrilebs, romlis mixedviTac rS dardeba mniSvnelovnebis donis

kritikul c mniSvnelobas da niSanTa damoukideblobis H0 hipoTzas

uaryofen, Tuki | rS | > c.

amocanebi1. Seswavlil iqna 28 mozardis kbilis garsis anTebis (periodo-

ntitis) doneebs Soris sxvaoba dantisturi programis win da misi Ca-

tarebidan eqvsi Tvis Semdeg. aRmoCnda, rom 15 avadmyofis mdgomareobagaumjobesda, 8-isa gauaresda da 5-isa ucvleli darCa. ra SeiZleba

iTqvas programis dadebiT mxareze?

2. davuSvaT wina amocanaSi cvlileba aRiricxeba Svid balian

skalaSi (-3-dan 3-mde), sadac 3 da -3 Sesabamisad, niSnavs gaumjobese-

218

bisa da gauaresebis umaRles xarisxebs, 0 ki, – ucvlel statuss damiviReT Semdegi monacemebi:

cvlilebis xarisxi avadmyofTa raodenoba

+3 4+2 5+1 60 5-1 4-2 2-3 2

2.1. romeli statistikuri kriteriumi unda gamoviyenoT imisaTvis,

rom davadginoT mdgomareobis gaumjobesebis sakiTxi?

2.2. rogoria 2.1-is Sesabamisi kriteriumis p-mniSvneloba?3. davuSvaT, mocemulia ori amokrefa, romelTa moculobebia 6

da 7 da rangTa jami pirvel amokrefaSi tolia 58-is. uilkoksonis

rangTa jamebis kriteriumis gamoyenebiT daadgineT Sedegis mniSvnelov-

nebis sakiTxi.

4. cxovelebis qcevis Sesaswavlad Seqmnilia qcevis gazomvis me-

Todebi. gazomili iyo katebis TamaSisa da devnis qulebi. Ees qulebi

warmodgenilia cxrilSi:

TamaSi 2 6 8 4 7 5 10 3 1 9

devna 6.5 8 4 1 5 9 10 2 3 6.5

4.1. gamoTvaleT spirmenis korelaciis koeficienti;4.2. CamoayalibeT statistikuri hipoTeza, romelic saWiroa imisaTvis,

rom SevamowmoT, mniSvnelovnad gansxvavdeba Tu ara es koeficienti

nulisagan.

5. arsebobs azri, rom leqciebze daswreba da saboloo Sefaseba

korelaciaSia. aRricxuli iyo 12 studentis daswreba da gamocdaze

miRebuli qulebi:

Ggacdenebi 22 6 1 8 6 11 10 6 10 0 5 10

gamocdis qula 72 88 99 68 66 77 52 78 76 96 91 86

5.1. am monacemebisaTvis ipoveT korelaciis koeficienti;

5.2. SeamowmeT mniSnelovnad gansxvavdeba Tu ara es koeficienti nuli-sagan;

5.3. ra daskvnas gaakeTebT gacdenebisa da gamocdis qulebs Soris kavS-

irze.

6. SeiZleba Tu ara viwinaswarmetyveloT saboloo Sefaseba Sua-

leduri gamocdis Sefasebis mixedviT?

Sualeduri qula 87 89 97 80 73 85 81 74 80 89

saboloo qula 84 91 96 87 66 90 79 80 89 86

6.1. daxazeT gabnevis diagrama;

219

6.2. SeamowmeT mniSvnelovnad gansxvavdeba Tu ara korelaciis koefici-enti nulisagan.

6.3. ra daskvnas gaakeTebT qulebis kavSiris Sesaxeb?

7. rogoria kavSiri fizikur momxibvlelobasa da popularobas

Soris sabavSvo baRis aRsazrdel bavSvebSi? 12 gogonasa da 14 biWis

Sefasebis Sedegad miiRes Semdegi rangebi:

Ggogonebi

fizikuri mimzidveloba 3 6 2 9 12 1 7 8 4 10 5 11

popularoba 1 7 3 10 11 3 8 7 5 9 6 12

biWebi

mimzidveloba 5 11 3 9 4 8 12 1 14 10 6 13 2 7

popularoba 9 1 10 4 6 12 5 13 8 14 2 11 7 3

7.1. ra daskvnas akeTebT gogonebSi momxibvlelobasa da popularobis

rangebs Soris kavSiris Sesaxeb?

7.2. ra daskvnas akeTebT biWebSi momxibvlelobasa da popularobis ra-

ngebs Soris kavSiris Sesaxeb?

8. ra gavlenas axdens kompiuteris ekranis xangrZlivad yureba

aRqmis unarze?

dro (sT) 0.5 1.7 4.2 6.0 2.8 5.3 0.9

aRqma (sm) 4.3 3.1 2.1 0.4 6.3 3.7 1.6

SerCevis safuZvelze ra daskvnas akeTebT kompiuterTan muSaobis saaT-

ebsa da siRrmis aRqmis Secdomas Soris kavSiris Sesaxeb?

9. mozardebis skoliT kmayofilebasTan dakavSirebuli faqtore-

bis Sesaswavlad, gamokiTxuli iyo 10 moswavle. erT-erTi cvladi iyo

skolasTan dakavSirebuli yoveldRiuri uaryofiTi movlenebis raoden-

oba.uaryofiTi movlenebi 7 14 8 19 6 10 17 7 12 17

skoliT kmayofileba 34 14 24 15 29 27 14 38 19 21

ra daskvnas akeTebT uaryofiTi movlenebis raodenobasa da skoliT

kmayofilebas Soris kavSiris Sesaxeb?

10. maswavlebelma Seamowma sakontrolos formis gavlena mosw-

avleTa codnaze. gavlena gamoixateboda moswavleebis mier sakontro-

lo gamokiTxvaze miRebul niSnebSi. amisaTvis, maswavlebelma erTidai-

mave masalis gamoyenebiT Seadgina sakontrolo samuSao da testi. mox-

da moswavleebis daxarisxeba ori kategoriis mixedviT: Seasrula dava-

leba, ver Seasrula davaleba.sakontrolo

Seasrula ver Seasrula

Seasrula 6 2 21

testi

ver Seasrula 4 12

romeli kriteriumi unda gamoiyenoT Sesamowmeblad?

220

11. Catarda eqsperimenti, raTa SeemowmebinaT garkveul saxelmZ-RvaneloSi arsebuli amocanaTa sistemis mizanmimarTuleba. am mizniT,

mocemul savarjiSoebTan dakavSirebiT, Seiswavles maswavebelTa azri

saswavlo wlis dasawyisSi da wlis bolos. imis gamo, rom maswavleb-

elTa azrebi aRmoCnda Zlier gansxvavebuli (maswavlebelTa erTi nawi-

li iTxovda savarjiSoebis raodenobis gazrdas, meore nawili ki _

Semcirebas), gadawyvites daxarisxebinaT gamokiTxvis Sedegebi or kate-

goriaSi: “ gaizardos” da “ Semcirdes”.

Ppirveli azri gaizardos gaizardos Semcirdes Semcirdes

Mmeore azri gaizardos Semcirdes gaizardos Semcirdes masw. raod. 52 74 36 38

icvleba Tu ara maswavleblebis Sexeduleba wlis ganmavlobaSi?

12. iTvleba, rom maTematikis kursis Seswavla, xels uwyobs mo-

swavleebSi logikuri azrovnebis ganviTarebas, maSinac ki roca aRniSn-

uli azrovnebis formireba ar xdeba mizanmimarTulad. Semowmebis miz-

niT, orjer Catarda sakontrolo samuSao: seqtembris bolos da mais-

is bolos. aRniSnuli sakontroloebi dawera sxvadasxva skolidan Sem-

TxveviT SerCeulma 35-ma moswavlem. sakontroloebSi arsebuli daval-

ebebis amoxsna dafuZnebuli iyo logikuri azrovnebis unaris gamoyene-

baze. iyo SemuSavebuli Semdegi Sefaseba: sworad amoxsnili 1 an 2 am-ocana _ qula “0”, sworad amoxsnili 3 amocana _ qula “1”, sworad

amoxsnili 4 amocana _ qula”2’, sworad amoxsnili 5 amocana _ qula

“3”.

pirveli Sefaseba 1 1 1 2 0 0 0 0 0 1 0 3 2 1 0 0 1 0 1 0 0 1 2 0 0

meore Sefaseba 1 0 0 2 1 0 1 0 0 1 1 3 2 0 1 1 1 1 1 1 0 2 3 1

maTematikis Seswavla uwyobs xels Tu ara logikuri azrovnebis Camo-

yalibebas?

13. eqsperimentatorebma Seiswavles Tu ra gavlenas axdens gark-

veuli filmis yureba, moswavlis mier garkveuli masalis aTvisebaze.

moswavleebis mier aTvisebis donis Sesamowmeblad Caatares sakontro-loebi, romelic gaTvlili iyo saSualo moswavleze. aRniSnuli sakon-

troloebi Catarda erTsa da imave 12 moswavleze, mocemuli filmis

yurebamde da filmis yurebis Semdeg.

swori pasuxebis raodenoba

I Semowmeba 3 8 4 5 4 5 6 7

II Semowmeba 4 8 6 6 2 6 10 7

SeamowmeT hipoTeza: filmis yureba ar cvlis masalis aTvisebas. gamoi-yeneT vilkoksonis kriteriumi.

13. erTerTi qalaqis or skolaSi Seamowmes me_8 klaselTa mi-

er, savarjiSoebis Sesrulebis unari. Semowmeba moxda specialurad Se-

dgenili savarjiSoebis daxmarebiT. Mmaqsimaluri qula _ 10, minimalu-

221

ri _ 1. orive skolaSi daTvlili iyo sixSireebi miRebuli qulebisSesabamisad.

qulebi 10 9 8 7 6 5 4 3 2 1

I skola 2 1 3 4 6 5 4 1 2 -

II skola 1 2 3 7 3 5 3 - 2 -

medianuri kriteriumis gamoyenebiT SeamowmeT gansxvavdeba Tu ara erT-

maneTisagan sxvadasxva skolis moswavleTa unarebis done.

14. Catarebuli iyo eqsperimenti saukeTeso saxelmZRvanelos ga-movlenis mizniT. anketis erTi kiTxva iyo Semdegi: ramdenad SesaZlebe-

lia mocemuli saxelmZRvaneloTi aiTviso is masala, romelic maswav-

lebels ar auxsnia (savaraudo pasuxebia: ki _ ara). Aaseve moxda, im

weriTi namuSevrebis Sedegebis Sedareba, romelTa daxmarebiTac Semowm-

da mocemuli kursis erT-erTi nawilis aTvisebis done.

Kki ara

skola # 1 15 5

skola # 2 7 8

gansxvdeba Tu ara ori skolis maswavlebelTa azri erTmaneTisagan?

222

danarTi (statistikuri cxrilebi)

puasonis ganawilebis cxrilebi ( ( )!

k

P k ek

)

= 1.0 = 1.5 = 2.0 = 2.5 = 3.0 = 3.5 = 4.0 =4.5 = 5.0

p(0) 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067p(1) 0.3679 0.3347 0.2707 0.2052 0.1494 0.1057 0.0733 0.0500 0.0337p(2) 0.1839 0.2510 0.2707 0.2565 0.2240 0.1850 0.1465 0.1125 0.0842p(3) 0.0613 0.1255 0.1804 0.2138 0.2240 0.2158 0.1954 0.1687 0.1404p(4) 0.0153 0.0471 0.0902 0.1336 0.1680 0.1888 0.1954 0.1898 0.1755p(5) 0.0031 0.0141 0.0361 0.0668 0.1008 0.1322 0.1563 0.1708 0.1755p(6) 0.0005 0.0035 0.0120 0.0278 0.0504 0.0771 0.1042 0.1281 0.1462p(7) 0.0001 0.0008 0.0034 0.0099 0.0216 0.0385 0.0595 0.0824 0.1044p(8) 0.0001 0.0009 0.0031 0.0081 0.0169 0.0298 0.0463 0.0653

p(9) 0.0002 0.0009 0.0027 0.0066 0.0132 0.0232 0.0363

p(10) 0.0002 0.0008 0.0023 0.0053 0.0104 0.0181

p(11) 0.0002 0.0007 0.0019 0.0043 0.0082

p(12) 0.0001 0.0002 0.0006 0.0016 0.0034

p(13) 0.0001 0.0002 0.0006 0.0013

p(14) 0.0001 0.0002 0.0005

p(15) 0.0001 0.0002

standartuli normaluri ganawilebis zeda -kritikuli

wertilebi ( z )

0.1 0.05 0.025 0.125 0.01 0.005 0.0025 0.001z 1.28 1.64 1.96 2.24 2.33 2.57 2.81 3.08

223

(0.1)N -is simkvrivis (2 / 21

( )2

zz e

) mniSvnelobebi

224

(0.1)N -is ganawilebis funqciis (

xt

dtex 2

2

2

1)( ) mniSvnelobebi

225

2

20

0

1( )

2

z t

z e dt

funqciis cxrilebi

226

t (stiudentis) ganawilebis zeda -kritikuli wertilebi ( ,nt )

n 0.1 0.05 0.025 0.01 0.005 0.0025 0.001

1 3.078 6.314 12.706 31.821 63.656 127.321 318.289

2 1.886 2.920 4.303 6.965 9.925 14.089 22.328

3 1.638 2.353 3.182 4.541 5.841 7.453 10.214

4 1.533 2.132 2.776 3.747 4.604 5.598 7.173

5 1.476 2.015 2.571 3.365 4.032 4.773 5.894

6 1.440 1.943 2.447 3.143 3.707 4.317 5.208

7 1.415 1.895 2.365 2.998 3.499 4.029 4.785

8 1.397 1.860 2.306 2.896 3.355 3.833 4.501

9 1.383 1.833 2.262 2.821 3.250 3.690 4.297

10 1.372 1.812 2.228 2.764 3.169 3.581 4.144

11 1.363 1.796 2.201 2.718 3.106 3.497 4.025

12 1.356 1.782 2.179 2.681 3.055 3.428 3.930

13 1.350 1.771 2.160 2.650 3.012 3.372 3.852

14 1.345 1.761 2.145 2.624 2.977 3.326 3.787

15 1.341 1.753 2.131 2.602 2.947 3.286 3.733

16 1.337 1.746 2.120 2.583 2.921 3.252 3.686

17 1.333 1.740 2.110 2.567 2.898 3.222 3.646

18 1.330 1.734 2.101 2.552 2.878 3.197 3.610

19 1.328 1.729 2.093 2.539 2.861 3.174 3.579

20 1.325 1.725 2.086 2.528 2.845 3.153 3.552

21 1.323 1.721 2.080 2.518 2.831 3.135 3.527

22 1.321 1.717 2.074 2.508 2.819 3.119 3.505

23 1.319 1.714 2.069 2.500 2.807 3.104 3.485

24 1.318 1.711 2.064 2.492 2.797 3.091 3.467

25 1.316 1.708 2.060 2.485 2.787 3.078 3.450

26 1.315 1.706 2.056 2.479 2.779 3.067 3.435

27 1.314 1.703 2.052 2.473 2.771 3.057 3.421

28 1.313 1.701 2.048 2.467 2.763 3.047 3.408

29 1.311 1.699 2.045 2.462 2.756 3.038 3.396

30 1.310 1.697 2.042 2.457 2.750 3.030 3.385

227

t ganawilebis zeda / 2 -kritikuli wertilebi , / 2nt (orkudiani)

n A0.800.20

0.900.10

0.950.05

0.980.02

0.990.01

0.9950.005

0.9980.002

0.9990.001

1 3.0786.31412.70631.82063.657127.321318.309636.6192 1.8862.920 4.303 6.965 9.925 14.089 22.327 31.5993 1.6382.353 3.182 4.541 5.841 7.453 10.215 12.9244 1.5332.132 2.776 3.747 4.604 5.598 7.173 8.6105 1.4762.015 2.571 3.365 4.032 4.773 5.893 6.8696 1.4401.943 2.447 3.143 3.707 4.317 5.208 5.9597 1.4151.895 2.365 2.998 3.499 4.029 4.785 5.4088 1.3971.860 2.306 2.897 3.355 3.833 4.501 5.0419 1.3831.833 2.262 2.821 3.250 3.690 4.297 4.78110 1.3721.812 2.228 2.764 3.169 3.581 4.144 4.58711 1.3631.796 2.201 2.718 3.106 3.497 4.025 4.43712 1.3561.782 2.179 2.681 3.055 3.428 3.930 4.31813 1.3501.771 2.160 2.650 3.012 3.372 3.852 4.22114 1.3451.761 2.145 2.625 2.977 3.326 3.787 4.14015 1.3411.753 2.131 2.602 2.947 3.286 3.733 4.07316 1.3371.746 2.120 2.584 2.921 3.252 3.686 4.01517 1.3331.740 2.110 2.567 2.898 3.222 3.646 3.96518 1.3301.734 2.101 2.552 2.878 3.197 3.610 3.92219 1.3281.729 2.093 2.539 2.861 3.174 3.579 3.88320 1.3251.725 2.086 2.528 2.845 3.153 3.552 3.85021 1.3231.721 2.080 2.518 2.831 3.135 3.527 3.81922 1.3211.717 2.074 2.508 2.819 3.119 3.505 3.79223 1.3191.714 2.069 2.500 2.807 3.104 3.485 3.76824 1.3181.711 2.064 2.492 2.797 3.090 3.467 3.74525 1.3161.708 2.060 2.485 2.787 3.078 3.450 3.72526 1.3151.706 2.056 2.479 2.779 3.067 3.435 3.70727 1.3141.703 2.052 2.473 2.771 3.057 3.421 3.69028 1.3131.701 2.048 2.467 2.763 3.047 3.408 3.67429 1.3111.699 2.045 2.462 2.756 3.038 3.396 3.65930 1.3101.697 2.042 2.457 2.750 3.030 3.385 3.646

1.2821.645 1.960 2.326 2.576 2.807 3.090 3.291

228

2 (xi kvadrat) ganawilebis zeda -kritikuli wertilebi ( 2,n )

229

( , )F n m (fiSeris) ganawilebis zeda -kritikuli wertilebi ( , ,n mF )

230

( , )F n m (fiSeris) ganawilebis zeda -kritikuli wertilebi ( , ,n mF )

231

Iv. Javakhishvili Tbilisi State UniversityFaculty of Exact and Natural Sciences

Syllabus

Title of theCourse

Probability and Mathematical Statistics for Chemistry,Biology, and Life Sciences

Code of thecurseStatus of thecourse

The obligatory one term is provided for the undergraduatestudents of Faculty of Biology and Life Sciences

ECTS 6 credits: 60 contact hours (lecture – 30, practical works– 30,Laboratory works) 90 hours for the independent work

Lecturers Prof. Omar Purtukhia, Iv. Javakhishvili Tbilisi State Univer-sity, Faculty of Exact and Natural Sciences, Phone: 304145(office), 189346 (home), 899503082 (mobil) e-mail:[email protected]; [email protected]. Zurab Tsigroshvili, Georgian Tekhnical University,Georgian-American University, Phone: 304145 (office),899317024 (mobil), e-mail: [email protected] Manjgaladze, invited lecturer, Iv. JavakhishviliTbilisi State University, Faculty of Exact and Natural Scie-nces, Phone: 304145 (office), 968771 (home), e-mail:[email protected]

The aim ofthe course

The aim of this lecture-notes is to give the students of Chem-istry, Biology and Life sciences the skills for working withreal data coming from corresponding field.

Pre-request Advanced Mathematics, Part I – CalculusFormat of thecourse

Lecture, Practical work, Laboratory work

contents ofthe course

Lecture1. Subject of statistics. Population and sample. Pro-blems of Descriptive and Inferential Statistics. Types of Da-ta. Graphical representation of Data. Problems. (s. [1], Chap.I.1-I.3, I.6; [2], Chap. 2, § 1-2; [4], Chap. 1-2)

2h lecture, 2h practical workLecture2. Sample characteristics of Data. Measures of ce-ntral location. Measures of spread. Some properties of samp-le characteristics. Mathed observations. Correlation. Probl-ems. (s. [1], Chap. I.7-I.9; [2], Chap. 3, § 1-3; [4], Chap. 3)

2h lecture, 2h practical workLecture3. Elements of Probability Theory. Random exper-iment, Probability space, Random Events, Random Variabl-es. Definition of Probability. Properties of Probabilities. Co-nditional Probabilities. The Multiplicative Law of Probabili-





232

ty. Independence of the events. Complete Probability andBayes’s formula. Problems. (s. [1], Chap. II.1-II.2; [2], Chap.4, § 1-5; [3], Chap. I.1-I.2; [4], Chap. 5)

2h lecture, 2h practical workLecture4. Discrete Random Variables and Distributions.Probability Mass and Distribution Functions for a DiscreteRandom Variables. The expected value and variance of rand-om variables. Permutations and Combinations. The Binomialand Poisson Distributions and Their Connection. Problems.(s. [1], Chap. II.3-II.4; [2], Chap. 5, § 1-6; [3], Chap. I.3-I.4;[4], Chap. 6)

2h lecture, 2h practical workLecture5. Continuous Random Variables and Distributi-ons. Continuous random variables and their distribution. De-nsity function. Normal Random Variables. Approximationsof Binomial and Poisson Distributions. Problems. (s. [1],Chap. II.5-II.6; [2], Chap. 6, § 1-4; [3], Chap. I.5-I.7; [4],Chap. 7)

2h lecture, 2h practical workLecture6. Estimation Theory. The Point Estimation. Thepoint estimators. Central Limit Theorem. Estimation for theBinomial Distribution. Estimation for the Poisson Distributi-on. Problems. (s. [1], Chap. III.5-III.4; [2], Chap. 7, § 1-5;[3], Chap. II.1)

2h lecture, 2h practical workLecture7. Estimation Theory. The Interval Estimation.Estimation of the Population Mean and Variance. Interval es-timation of the variance of the Normal population. IntervalEstimation for the Binomial Distribution. Interval Estimationfor the Poisson Distribution. Problems. (s. [1], Chap. III.5;[2], Chap. 8, § 1-4; [3], Chap. II.1-II.3; [4], Chap. 8)

2h lecture, 2h practical workLecture8. Hypothesis Testing: One-sample Inference.One-sample Test for the Mean of a Normal Distribution withKnown Variance. One-sample Test for the Mean of a Norm-al Distribution with Unknown Variance. The Power Calcula-tion. Sample-size determination. One-sample Test for theVariance of a Normal Distribution. Hypothesis Testing forthe Binomial Distribution. Hypothesis Testing for the Poiss-on Distribution. Problems. (s. [1], Chap. III.6; [2], Chap. 9, §1-8; [3], Chap. II.4-II.8; [4], Chap. 9)

2h lecture, 2h practical workLecture9. Hypothesis Testing: Two-sample Inference.The Paired t Test. Two-sample t Test for the Normal Meanswith Equal Unknown Variances. Test for equality of the No-

233

rmal Variances. Two-sample t Test for the Normal Meanswith Unequal Variances. Sample-size determination. Probl-ems. (s. [1], Chap. III.7; [2], Chap. 10, § 1-6; [3], Chap. II.9-II.10; [4], Chap. 10)

2h lecture, 2h practical workLecture10. Multi-sample Inference Analysis of Variance(ANOVA). Hypothesis Testing in One-way ANOVA modelwith fixed effects. Comparisons of specific groups in One-way ANOVA. Linear contrasts. Multiple Comparisons (Bon-feroni’s and Sheppe’s Methods). Hypothesis Testing in One-way ANOVA model with random effects. Problems. (s. [1],Chap. III.8; [2], Chap. 11, § 1-6; [3], Chap. II.19; [4], Chap.12)

2h lecture, 2h practical workLecture11. Two-sample Problems for Binomial Proporti-ons. Categorical Data. Comparison of two binomial propor-tions – Normal approximation. Comparison of two binomialproportions – 22 Contingency Table. 22 Contingency Tab-le for independence. Problems. (s. [1], Chap. III.9; [2], Chap.12, § 1-2; [3], Chap. II.11; [4], Chap. 10)

2h lecture, 2h practical workLecture12. Categorical Data. Measures of Effect for Categ-orical Data. Fisher’s exact Test. Two-sample Test for Bino-mial Proportions for Matched-Pair Data – McNemar’s Test.Problems. (s. [1], Chap. III.9; [2], Chap. 12, § 3-5; [3], Chap.II.9-II.11; [4], Chap. 10)

2h lecture, 2h practical workLecture13. Categorical Data. Estimation of Sample Sizeand Power for Comparing of Two Binomial Proportions. rcContingency Tables. 2-Goodness-of-fit Test. Problems. (s.[1], Chap. III.10-III.11; [2], Chap. 12, § 6-8; [3], Chap. II.14-II.16; [4], Chap. 12)

2h lecture, 2h practical workLecture14. Regression and Correlation Methods. Generalconcepts. Fitting Regression Lines – The Method of LeastSquares. Inference About Parameters from Regression Lines.Interval Estimation for Linear Regression. The CorrelationCoefficient. Statistical Inference for Correlation Coefficients.Partial and Multiple correlation. Multiple Logistic Regressi-on. Problems. (s. [1], Chap. III.13; [2], Chap. 13, § 1-11; [3],Chap. II.17-II.18; [4], Chap. 11)

2h lecture, 2h practical workLecture15. Nonparametric Methods. The Sign Test. TheWilcoxon Signed-Rank Test. The Wilcoxon Rank-Sum Test.

234

The Kruskal-Wallis Test. Spearman’s Rank Correlation Coe-fficient. Problems. (s. [1], Chap. III.12; [2], Chap. 14, § 1-7;[3], Chap. II.18)

2h lecture, 2h practical work

References1. O. Purtukhia. Descriptive Statistics, Probability and Infer-ential Statistics. Tbilisi, 2008.2. B. Rosner. Fundamentals of Biostatistics. Published byDuxbury, 1995.3. O. Purtukhia. Probability and Statistics in Examples andProblems. Tbilisi, 2009.4. Allan G. Bluman. Ementary Statistics: a brief version, sec-ond edition. Published by McGraw-Hill, New York, 2003.

Grades 100 points grades are used:1. two written tutorials with three questions each up to fivepoints;2. students activity at practical and laboratory works up to20 points;3. attendance at lectures and practical works up to 10 points;4. final written exam with four questions each up to 10points.

Examprerequest

Within the first three parameters of grades students have toearn at least 30 points and to take part at least at one tutorial.

Gradingscheme

Attendance 10%Participation in tutorials (2x15) 30%Activities at practical (15%) and laboratoryworks (5%)

20%

Final exam 40%Final grade 100%

Obligatoryliterature

1. O. Purtukhia. Descriptive Statistics, Probability and Infer-ential Statistics (in Georgian). Tbilisi, 2008.2. B. Rosner. Fundamentals of Biostatistics. Published byDuxbury, 1995.3. O. Purtukhia. Probability and Statistics in Examples andProblems (in Georgian). Tbilisi, 2009.4. Allan G. Bluman. Ementary Statistics: a brief version, se-cond edition. Published by McGraw-Hill, New York, 2003.

Additionalliterature

1. G. Mania. Probability Theory and Mathematical Statistics(in Georgian). TSU, Tbilisi, 1976.2. N. Lazrieva, M. Mania, G. Mari, A. Mosidze, A. Toronja-dze, T. Toronjadze, T. Shervashidze. Probability Theory andMathematical Statistics for Economics (in Georgian). Found-

235

ation of “ Evrazia”, Tbilisi, 2000.3. E. Nadaraya, R. Absava, M. Patsatsia. Probability Theory(in Georgian). Tbilisi, 2009.4. O. Purtukhia. Probability Theory and Mathematical Stati-stics (in Georgian). Tbilisi, 2007.5. P. Newbold, W. L. Carlson, B. M. Thorne. Statistics forBusiness and Economics, sixth edition. Prentice Hall, UpperSaddle River, New Jersey, 2007.6. V. Feller. Introduction in Probability Theory and Applicat-ion,. vol. I, II (in Russian). Moscow, 1967.

Results ofstudy

As a result we expect that based on the statistical tools stud-ents could derive analysis of their data and could derive cor-rect conclusions.

236

sagnobrivi saZiebeli

albaToba, 15albaTobaTa gamoklebis wesi, 52albaTobaTa gamravlebis wesi,

53, 54albaTobaTa Sekrebis wesi, 51, 52albaTobis aqsiomaturi safuZve-

li, 51albaTurad damoukidebeli, 54albaTuri sivrce, 47alternatiuli hipoTeza, 104amovardnili monacemebi (outlier),

33amokrefa, 19aposterioruli albaToba, 57aprioruli albaToba, 57arabalansirebuli SemTxveva, 147araparametruli hipoTeza, 104ar aris kavSiri, 39arawrfivi kavSiri, 39aRweriTi statistika, 15atributuli monacemebi, 20aucilebeli xdomiloba, 49baiesis formula, 57balansirebuli SemTxveva, 147beitsonis magaliTi, 157berens-fiSeris problema, 126bernulis SemTxveviT sidide, 91bimodaluri SerCeva, 31binomuri ganawileba, 67bonferonis statistikuri kri-

teriumi, 145boqsploti (boxplot), 33gabnevis diagrama, 39gadanacvleba, 65gadauadgilebadi (Caunacvlebe-

li) Sefaseba, 87gadaxra jgufis SigniT, 137gaerTianeba, 48gamoZaxili (prognozi), 187wertilovani Sefasebebi, 85

ganawilebis kanoni, 61ganawilebis mediana, 210ganawilebis simkvrive, 74ganawilebis funqcia, 64, 74ganSrevebuli SemTxveviTi SerCe-

va, 16gafantulobis diagrama, 187dadebiTad korelirebuli, 199dadebiTad wanacvlebuli (asime-triuli), 31

dadebiTi wrfivi kavSiri, 39dakvirvebuli SeuRlebis cxri-

li, 155, 177damatebiTi xdomiloba, 48damoukidebeli SemTxveviTi sid-

ideebi, 63determinaciis koeficienti, 193decili (decile), 32did ricxvTa kanoni, 5, 88diskordantuli wyvili, 170diskretizacia, 23diskretuli albaTuri sivrce,

47diskretuli monacemebi, 20diskretuli SemTxveviTi sidi-

de, 60dispersia, 63elementaruli xdomiloba, 47empiriuli ganawilebis funqcia,

21erTfaqtoriani ANOVA modeli,

136erTfaqtoriani dispersiuli an-

alizi, 135efeqturi Sefaseba, 89F-kriteriumi, 125varianti, 22, 29variaciuli mwkrivi, 21venis diagramebi, 49vulfis meTodi, 165

237

zrdadobis Tviseba, 51zusti meTodi, 211Tavisuflebis xarisxi, 96Tanabari ganawileba, 74TanakveTa, 48Tanxmobis amocana, 182Tanxmobis F–kriteriumi, 192Tanxmobis t-kriteriumi., 193Tanxmobis 2 kriteriumi, 182kategoria (klasi), 23kvartilTSorisi gabnevis diapa-

zoni (Interquartial range), 32klasteruli SerCeva, 16kombinatorikis elementebi, 66kovariaciis koeficienti, 199konkordantuli wyvili, 170korelaciis Teoria, 186korelaciis kerZo koeficienti,

204korelaciis koeficienti, 40, 199kraskel-uolisis kriteriumi,

219kriteriumis mniSvnelovnebis

done, 106kriteriumis simZlavre, 106, 110logisturi gardaqmna, 204logisturi regresia, 186logisturi regresiis modeli,

204maTematikuri lodini, 62, 75maTematikuri statistika, 6maknemaris kriteriumi, 170man-uitnis kriteriumi, 217marginaluri sveti, 154marginaluri striqoni, 154martivi SemTxveviTi SerCeva, 85martivi SerCeva, 16marjvniv asimetriuli ganawile-

ba, 32mendelis kanoni, 158II gvaris Secdomis albaToba,

105

meore kvartili, 32mesame kvartili, 32misadagebis wiri, 39, 40mosalodneli SeuRlebis cxri-

li, 155, 177mravlobiTi korelaciis koefi-

cienti, 204mravlobiTi Sedarebis kriteri-

umi, 144naSTTa analizi martivi wrfivi

regresiisaTvis, 197naSTiTi komponenti (wevri), 190naSTiT wevrTa kvadratebis ja-

mi, 191ndobis albaToba, 95ndobis intervali, 94ndobis intervalebi regresiis

parametrebisaTvis, 194ndobis intervalebi regresiis

prognozebisaTvis, 195niSanTa damoukidebloba, 154niSanTa damoukideblobis

amocana, 178niSnebis kriteriumi, 209nominaluri tipis monacemebi, 20normaluri SemTxveviTi sidide,

76nulovani hipoTeza, 104oramokrefiani amocana, 120ordinaluri skala, 212ordinaluri tipis monacemebi,

20ormxrivi alternativa, 106parametruli hipoTeza, 104p-mniSvneloba, 114p-rigis kvantili, 79p rigis procentili, 32I gvaris Secdomis albaToba, 105pirveli kvartili (quartile), 32pirobiTi albaToba, 53pirobiTi maTematikuri lodini,

187

238

placebo, 17, 19poligoni, 25populacia, 16populaciaTa erTgvarovnebis am-

ocana, 178populaciis SerCeva (amorCeva),

15procentili, 32procentilebi (percentiles), 32prognozireba, 187proporciaTa erTgvarovnebis am-

ocana, 154procentuli rangi, 33puasonis ganawileba, 69rangi, 21, 33rangobrivi kriteriumebi, 209regresiis komponentTa kvadrat-

ebis jami, 191regresiis komponenti, 190regresiis funqcia (wiri), 187regresiis wrfe, 186regresiuli analizi, 40, 186regresiuli analizis amocana,

187regresiuli modelebi, 187rekursiuli (rekurentuli) ga-

daTvlis wesi, 68, 70reprezentatuli amokrefa, 19retrospeqtiuli gamokvleva,

166riskebis sxvaoba, 163satertvaitis meTodi, 126simetriuli SerCeva, 31sistematuri SerCeva, 16sixSire, 22, 29sixSireTa mdgradoba, 4spirmenis rangobrivi korelaci-

is koeficienti, 223sruli albaTobis formula, 56sruli (mTeli) kvadratebis ja-

mi, 191standartizacia, 78

standartuli gadaxra, 63standartuli normaluri ganaw-

ileba, 78standartuli normaluri cxri-

lebi, 78statistikis araparametruli

meTodebi, 208statistikurad mniSvnelovani,

114statistikurad umniSvnelo, 114statistikuri daskvnebi, 15statistikuri daskvnebis Teo-ria, 15, 17

statistikuri mdgradoba, 4statistikuri kriteriumi, 107,

113, 121, 123, 138, 142, 148statistikuri hipoTeza, 104stacionaruli procesi, 69stiudentis ganawileba, 96t -ganawileba Tavisuflebis

xarisxiT n, 96t –kriteriumi, 125, 135t-kriteriumis meTodologia, 135trendi, 181trimodaluri SerCeva, 31ualbaTesi ricxvi, 67uaryofiTad korelirebuli, 199uaryofiTad wanacvlebuli (asi-

metriuli) SerCeva, 31uaryofiTi wrfivi kavSiri, 39uTavsebadi, 47uilkoksonis niSniani rangebis

kriteriumi, 211uilkoksonis rangTa jamis kri-

teriumi, 214umcires kvadratTa meTodi, 40,

188unimodaluri SerCeva, 31ucnobi parametris miaxloebiTi

mniSvneloba, 87ucnobi parametris Sefaseba, 87uwyveti albaTuri sivrce, 47

239

uwyveti ganawileba, 74uwyveti tipis monacemebi, 20fardobiT riski, 163fardobiTi sixSire, 22fiSeris ganawileba, 124fiSeris zusti kriteriumi, 166fiSeris kriteriumi, 167foTlebiani Reroebis msgavsi

diagrama, 25qvesimravle, 47SemTxveviT efeqtebiani, erTfaq-toriani ANOVA modeli, 147

SemTxveviTi eqsperimenti, 4, 46SemTxveviTi sidide, 46, 60SemTxveviTi sididis standarti-

zacia, 78SemTxveviTi SerCeva, 16SemTxveviTi SerCevis meTodi, 19SemTxveviTi xdomiloba, 46SerCeva, 16SerCeviTi absoluturi gadaxra,

34SerCeviTi dispersia, 34, 85SerCeviTi variaciis koeficien-

ti, 36SerCeviTi kovariaciis koeficie-

nti, 190SerCeviTi mediana, 30SerCeviTi moda, 31SerCeviTi saSualo, 29, 85, 107SerCeviTi standartuli gadax-

ra, 34SerCevis diapazoni, 24, 31Sesworebuli standartuli ga-

daxra, 34Sesworebuli SerCeviTi disper-

sia, 34SeuRlebis r c cxrilebi, 176SeuRlebis cxrilis meTodi, 154SeuZlebeli xdomiloba, 48Sefaseba, 50

SefasebaTa efeqturobis Tviseba,88

Sefasebis standartuli Secdo-

ma, 88Sefasebis sqema, 87Sefes statistikuri kriteriu-

mi, 146Sidajgufuri variacia, 191Canacvleba, 90calmxrivi alternativa, 106centraluri zRvariTi Teorema,

90Zaldebuli (Zalmosili) Sefase-ba, 87

warmomadgenlobiTi amokrefa, 19wrfivi kontrasti, 141wrfivi prediqtori, 205wrfivi regresia, 185, 186xdomiloba, 46xdomilobaTa gaerTianeba, 48xdomilobaTa namravli, 49xdomilobaTa sruli sistema, 56xdomilobaTa sxvaoba, 49xdomilobaTa jami, 49xdomilobis albaToba, 5, 50xdomilobis indikatori, 210xi kvadrat ganawilebis kanoni

Tavisuflebis xarisxiT n, 97jgufTaSoris gadaxris sazomi,

137jgufTaSorisi variacia, 191jgufTaSorisi kvadratebis saS-

ualo, 138jgufis SigniT kvadratebissaSualo, 138

jufdeba, 65hipergeometriuli ganawileba,

166hipoTezaTa Semowmeba, 17hipoTezis Semowmebis zusti me-

Todi, 114histograma, 25

s a r C e v i

winasityvaoba ............................................................................................................... 3

Sesavali .............................................................................................................…………… 4

silabusi ......................................................................................................................... 7

I. aRweriTi statistikal e q c i a 1. Tavi 1. statistikis sagani. populacia da

SerCeva. aRweriTi da daskvniTi statistikis amocanebi. ................. 13

Tavi 2. monacemebis pirveladi damuSaveba.

2.1. monacemebis tipebi. ............................................................................................

2.2. monacemebis grafikuli warmodgena. .......................................................

amocanebi .........................................................................................................................

19

19

25

l e q c i a 2. Tavi 3. ricxviT monacemTa SerCeviTi max-

asiaTeblebi. .....................................................................................................................3.1. saSualo yofaqcevis maxasiaTeblebi. ......................................................

3.2. SerCevis gafantulobis (ganfenilobis) maxasiaTeblebi. ........

3.3. SerCeviTi ricxviTi maxasiaTeblebis Tvisebebi. ...........................

3.4. dawyvilebuli monacemebi. korelacia. ...............................................

amocanebi .........................................................................................................................

2727

30

35

36

39

II. albaTobis Teorial e q c i a 3. Tavi 4. albaTobis Teoriis elementebi. .

4.1. SemTxveviTi eqsperimenti, albaTuri sivrce, xdomiloba,

SemTxveviTi sidide. .................................................................................................

4.2. moqmedebebi xdomilobebze. ........................................................................

4.3. xdomilobis albaToba, albaTobaTa Tvisebebi. ..............................

4.4. pirobiTi albaToba. albaTobaTa gamravlebis wesi. xdomil-

obaTa damoukidebloba. ..........................................................................................

4.5. sruli albaTobisa da baiesis formulebi. ....................................

amocanebi .........................................................................................................................

44

44

45

47

50

52

54

l e q c i a 4. Tavi 5. diskretuli SemTxveviTi sidid-

eebi. ZiriTadi albaTuri ganawilebebi. .......................................................

5.1. SemTxveviTi sidideebi da maTi ricxviTi maxasiaTeblebi. .....5.2. diskretuli SemTxveviTi sididis ganawilebis funqcia. .......

5.3. gadanacvlebebi da jufdebebi. .................................................................

5.4. binomuri ganawileba. .....................................................................................

5.5. puasonis ganawileba. ......................................................................................

5.6. kavSiri binomur da puasonis ganawilebebs Soris. ...................

amocanebi .........................................................................................................................

57

5760

61

63

65

67

68

l e q c i a 5. Tavi 6. uwyveti SemTxveviTi sidideebi.

ZiriTadi albaTuri ganawilebebi. ...................................................................

6.1. uwyveti SemTxveviTi sidideebi da maTi ganawileba. .................

6969

241

6.2. normaluri SemTxveviTi sidideebi (normaluri ganawileba).6.3. binomuri da puasonis ganawilebebis aproqsimacia normalu-

ri ganawilebiT. ..........................................................................................................

amocanebi .........................................................................................................................

72

7577

III. daskvniTi statistikal e q c i a 6. Tavi 7. SefasebaTa Teoria. wertilovani

Sefasebebi. ......................................................................................................................

7.1. Sesavali. ................................................................................................................7.2. wertilovani Sefasebebi. ............................................................................

7.3. centraluri zRvariTi Teorema. ............................................................

7.4. binomuri populaciis p parametris wertilovani Sefaseba.

7.5. puasonis populaciis parametris wertilovani Sefaseba.

amocanebi .........................................................................................................................

79798085868787

l e q c i a 7. Tavi 8. SefasebaTa Teoria. intervaluri

Sefasebebi. ......................................................................................................................

8.1. Sesavali. ................................................................................................................

8.2. populaciis saSualos intervaluri Sefasebebi. ........................8.3. normaluri populaciis dispersiis intervaluri Sefaseba. .

8.4. binomuri populaciis p parametris intervaluri Sefaseba.

8.5. puasonis ganawilebis parametris intervaluri Sefaseba.

amocanebi .........................................................................................................................

89899092949495

l e q c i a 8. Tavi 9. hipoTezaTa Semowmeba. erTamokr-

efiani amocanebi. ........................................................................................................

9.1. Sesavali, ZiriTadi cnebebi. ......................................................................

9.2. hipoTezis Semowmeba normaluri populaciis saSualosaTv-

is cnobili dispersiis dros. ..........................................................................9.3. hipoTezis Semowmeba normaluri populaciis saSualosaTv-

is ucnobi dispersiis dros. .............................................................................

9.4. kriteriumis simZlavris gamoTvla. ....................................................

9.5. SerCevis minimaluri moculobis gansazRvra. ...............................

9.6. hipoTezis Semowmeba normaluri populaciis dispersiisaT-

vis (ormxrivi alternativa). ............................................................................

9.7. hipoTezis Semowmeba binomuri populaciis p parametrisSesaxeb (ormxrivi alternativa). ...................................................................

9.8. hipoTezis Semowmeba puasonis populaciis parametrisSesaxeb (mcire moculobis SerCevebisaTvis). ..........................................

amocanebi .........................................................................................................................

9999

101

103104106

107

108

110110

l e q c i a 9. Tavi 10. hipoTezaTa Semowmeba. oramokr-

efiani amocanebi. ........................................................................................................

10.1. Sesavali. ..............................................................................................................

10.2. dawyvilebuli monacemebi. ........................................................................

115

115

115

242

10.3. oramokrefiani t -kriteriumi toli, ucnobi dispersiebisSemTxvevaSi. ...................................................................................................................

10.4. hipoTeza ori normaluri populaciis dispersiaTa tol-

obis Sesaxeb. ................................................................................................................

10.5. oramokrefiani t -kriteriumi aratoli dispersiebis SemT-

xvevaSi. .............................................................................................................................

10.6. SerCevaTa moculobebis gansazRvra. ori populaciis saS-

ualoebis Sedarebis kriteriumis simZlavre. .........................................

amocanebi .........................................................................................................................

117

119

120

122

124

l e q c i a 10. Tavi 11. mravalamokrefiani amocanebi.

dispersiuli analizi. ............................................................................................

11.1. Sesavali. ..............................................................................................................

11.2. hipoTezaTa Semowmeba erTfaqtorian ANOVA modelSi.deterministuli efeqtebis SemTxveva. .........................................................

11.3. jgufTa Sedareba erTfaqtorian ANOVA modelSi. wyvil-

Ta Sedarebis t -kriteriumi. ...............................................................................11.4. wrfivi kontrastebi. ...................................................................................

11.5. mravlobiTi Sedareba (bonferonisa da Sefes meTodebi). .....

11.6. hipoTezaTa Semowmeba erTfaqtorian ANOVA modelSi Se-

mTxveviTi efeqtebis dros. .................................................................................

amocanebi .........................................................................................................................

129129

131

133136138

141143

l e q c i a 11. Tavi 12. oramokrafiani amocanebi binom-

uri proporciebisaTvis. kategoruli monacemebi. .................................

12.1. Sesavali. ..............................................................................................................

12.2 ori binomuri proporciis Sedareba, normaluri aproqsima-cia. .....................................................................................................................................

12.3 ori binomuri proporciis Sedareba, SeuRlebis 22 cxri-li. ......................................................................................................................................

12.4. SeuRlebis 22 cxrili. niSanTa damoukidebloba. ..................

amocanebi .........................................................................................................................

145145

146

148151153

l e q c i a 12.12.5. kategoruli monacemebis efeqtebis sazomebi. .............................

12.6. fiSeris zusti kriteriumi. ...................................................................

12.7. maknemaris kriteriumi proporciebisaTvis dawyvilebul

monacemebSi. ...................................................................................................................

amocanebi .........................................................................................................................

156159

162165

l e q c i a 13.

12.8. SerCevis moculobis gansazRvra da kriteriumis simZlav-

re ori binomuri proporciebis Sedarebisas. .........................................

12.9. SeuRlebis r c cxrilebi. ..............................................................

12.10. Tanxmobis 2 kriteriumi. ............................................................…………

167170

176

243

amocanebi ......................................................................................................................... 178

l e q c i a 14. Tavi 13. regresiuli analizi da kore-

lacia. ...............................................................................................................................13.1. Sesavali, ZiriTadi cnebebi. .....................................................................

13.2. umcires kvadratTa meTodi. ....................................................................

13.3. regresiis wrfis parametrebis aRricxva. ......................................

13.4. Tanxmobis kriteriumebi regresiis wrfisaTvis. ........................

13.5. intervaluri Sefasebebi wrfivi regresiisaTvis. .....................

13.6. naSTTa analizi martivi wrfivi regresiisaTvis. .....................

13.7. korelaciis koeficienti. .........................................................................

3.8. kerZo da mravlobiTi korelacia. .........................................................

13.9. mravlobiTi logisturi regresia. .....................................................

amocanebi .........................................................................................................................

179179182183185187190191196197198

l e q c i a 15. Tavi 14. daskvniTi statistikis arapar-

ametruli meTodebi. ................................................................................................14.1. Sesavali. ..............................................................................................................

14.2. niSnebis kriteriumi. ...................................................................................

14.3. uilkoksonis niSniani rangebis kriteriumi. ................................

14.4. uilkoksonis rangTa jamis kriteriumi. .........................................

14.5. Tanadoba uilkoksonis rangTa jamis kriteriumsa da 2 -

kriteriums Soris. ...................................................................................................

14.6. kraskel-uolisis kriteriumi. ..............................................................14.7. spirmenis rangobrivi korelaciis koeficienti. .......................

amocanebi .........................................................................................................................

201201201204206

210211215217

d a n a r T i (statistikuri cxrilebi) ...............................................Syllabus. Probability and Mathematical Statistics for Chemistry, Biolo-gy, and Life Sciences. ..................................................................................................sagnobrivi saZiebeli ..............................................................................................

222

231

236

244

Omar Purtukhia,Zurab Tsigroshvili,

Qetevan Manjgaladze

Probability and MathematicalStatistics

(Lecture Course for Chemistry, Biology, and Life Sciences)

T S U

245

omar grigolis Ze furTuxia daibada 1959 wels walenjixis rai-onSi. warCinebiT daamTavra Tbilisis v.m. komarovis sax. fizika-maTema-

tikis skola-internati (1975 w.) da Tbilisis saxelmwifo universit-

etis meqanika-maTematikis fakul-

teti (1980 w.). 1981-1984 wleb-

Si iyo Tbilisis saxelmwifo un-

iversitetisa (xelmZRvaneli –

prof. g. mania) da moskovis lom-

onosovis sax. saxelmwifo unive-

rsitetis aspiranti (xelmZRvane-

li – prof. n. krilovi). 1984wels daicva sakandidato diser-

tacia lomonosovis universitet-

is specializebul samecniero sa-

bWoze, romelsac xelmZRvanelob-

da akad. a. kolmogorovi. 1984

wlidan muSaobs Tbilisis a. raz-

maZis maTematikis institutis

albaTobis Teoriisa da maTemati-

kuri statistikis ganyofilebaSi

da Tsu albaTobis Teoriisa damaTematikuri statistikis kaTed-

raze. kiTxulobs ZiriTad da specialur kursebs Tsu zust da sabune-

bismetyvelo mecnierebaTa, ekonomikisa da biznesisa da socialur da

politikur mecnierebaTa fakultetebze albaTobis Teoriisa da maTema-

tikuri statistikis sxvadasxva mimarTulebebiT. 1990 wlidan iyo

Tsu stoqasturi analizisa da statistikuri gadawyvetilebebis samec-

niero/kvleviTi laboratoriis ufrosi mecnier-TanamSromeli. 2006

wlidan aris Tsu profesori. koordinacias uwevs albaTobis Teoriisa

da maTematikuri statistikis swavlebas Tsu ekonomikisa da biznesis

da socialur da politikur mecnierebaTa fakultetebze. 2007-2008wlebSi iyo Tsu zust da sabunebismetyvelo mecnierebaTa fakultetis

maTematikis saswavlo-samecniero institutis direqtori. 2009 wlid-

an (specialuri xelSekrulebiT) xelmZRvanelobs sagnebis “statistika

I”-isa da “statistika II”-is swavlebas Tsu socialur da politikur

mecnierebaTa fakultetze. sxvadaxva dros miRebuli aqvs fond “evraz-

iis” (1996-1997ww.), INTAS-is (1997-1999ww.), saqarTvelos mecniere-

baTaa erovnuli akademiis (2000-2005ww.), saqarTvelos mecnierebisa

da ganaTlebis saministros (2005-2006ww.), da saqarTvelos erovnu-

li samecniero fondis (2006-2009ww.) grantebi. aris saqarTvelos

statistikuri asociaciis wevri.

246

zurab pavles Ze cigroSvili daibada 1963 wels TbilisSi. warC-inebiT daamTavra Tbilisis v. m. komarovis sax. fizika-maTematikis sko-

la-internati da Tbilisis saxelmwifo universitetis meqanika-maTemat-

ikis fakulteti. 1990 wlidan muSaobs Tbilisis a. razmaZis maTemati-

kis institutis albaTobis Teoriisa da maTematikuri statistikis ga-

nyofilebaSi. 1998 wels profesor

estate xmalaZis xelmZRvanelobiT

daicva sakandidato disertacia fi-

zika-maTematikis mecnierebaTa darg-

Si. wlebis ganmavlobaSi wakiTxuli

aqvs Teoriuli da gamoyenebiTi xa-siaTis leqciaTa kursebi albaTob-

asa da maTematikur statistikaSi

Tsu meqanika-maTematikisa da bio-

samedicino genetikis fakulteteb-

ze, aswavlida ESM-Tbilisis marT-

vis evropul skolaSi (1993-

1997ww.), 2006 wlidan dRemde ki-

Txulobs gamoyenebiTi statistik-

isa da riskis Teoriis kursebs sa-

qarTvelos teqnikur da qarTul-amerikul universitetebSi. muSobdaaqtuaris Tanamdebobaze sadazRvevo kompaniebSi I-holdingi (2001-2003

ww.) da “aldagi” (2004-2006ww.), 2007-2008 wlebSi analitikosis

Tanamdebobaze Ti-bi-si bankis saoperacio riskebis ganyofilebaSi. wle-

bis manZilze miRebuli aqvs saerTaSoriso da saqarTvelos mecniereba-

Ta akademiis samecniero grantebi. gakeTebuli aqvs samecniero moxsene-

bebi moskovis, varSavis, amstrdamis, eindhovenisa da madridis univers-

itetebSi. aris saqarTvelos statistikuri da aqtuaruli asociacie-

bis wevri.

247

qeTevan viqtoris asuli manjgalaZe daibada 1942 wels ruseTSi.daamTavra Tbilisis 66-e saSualo skola oqros medalze da Tbilisis

ivane javaxiSvilis saxelmwifo univer-

sitetis fizikis fakulteti warCineb-

iT. 1973 wels (prof. amiran toronja-

Zisa da revaz CitaSvilis xelmZRvanel-

obiT) daicva sakandidato disertacia

fizika-maTematikis mecnierebaTa dargSi.

aris Tsu zust da sabunebismetyvelo

mecnierebaTa fakultetis mowveuli

pedagogi, maswavlebelTa profesiuliganviTarebis programis treneri, alba-

Tobis Teoriisa da maTematikuri sta-

tistikis VII, VIII, IX da X klasebis

saxelmZRvaneloebis ("ras gvimaven mona-

cemebi", "movida luwi, movida kenti",

"maTematika 10") Tanaavtori. monawile-

obas iRebda proeqtebSi: "ekonomikuri barometri", "axali programis

danergva saSualo skolaSi", "trenerebis momzadeba saSualo skolis

maswavlebelTa gadasamzadeblad". kiTxulobs leqciebis kursebs Tsu

zust da sabunebismetyvelo mecnierebaTa fakultetis biologiis mima-rTulebaze da socialur da politikur mecnierebaTa fakultetze.

monawileobs programaSi: “trenerTa treningis programa maTematikasa

da misi swavlebis meTodikaSi”