13 Eng Maths Editied

8/9/2019 13 Eng Maths Editied

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THE KNOWLEDGE BASED TRAINING CENTRE

ENGINEERING MATHEMATICS

ASSIGNMENT

TITLE:

With the help of examples illustrate how statistics and probabilitiescan be used in your respective field (Quality Control and Safety

During Construction ) for batch production sampling and or qualitycontrol of production

NO NAME

1. RAMROOP Bhavish Kumar

2. NEERAYE Krishna Kumar


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LECTURERS NAME: Mr. M Madelon


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Contents

1.0 Qualit and !a"et Con#erns in Constru#tion..................................................................$

2.0 Or%ani&in% "or Qualit and !a"et ...................................................................................'$.0 (or) and Material !*e#i"i#ations....................................................................................+

'.0 ,otal Qualit Control.......................................................................................................-

+.0 Qualit Control !tatisti#al Methods............................................................................/

.0 !tatisti#al Qualit Control ith !am*lin% Attri utes................................................11

-.0 !tatisti#al Qualit Control ith !am*lin% aria les.................................................1-

3.0 !a"et ..............................................................................................................................2$

/.0 Re"eren#es......................................................................................................................210.0 4ootnotes......................................................................................................................2-

,a l

,a le 1 5 6oad Ratio and Pa 4a#tor..........................................................................................-,a le 25 Non"atal O##u*ational 7n8ur and 7llness 7n#iden#e Rates........................................2$,a le $5 4atal O##u*ational 7n8uries in Constru#tion9 1//- and 200'.....................................2'

,a le ' 5 4atalit Causes in Constru#tion9 1// :1//- and 200 :200-....................................2+

4i%uresY

4i%ure 15 E;am*le O*erative Chara#teristi# Curves 7ndi#atin% Pro a ilit o" 6ot A##e*tan#e..................................................................................................................................................1'4i%ure 2 5 aria le Pro a ilit


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1.0 Quality a ! Sa"#ty C$ %#& ' i C$ 't&u%ti$

Qualit #ontrol and sa"et re*resent in#reasin%l im*ortant #on#erns "or *ro8e#t mana%ers.

ood *ro8e#t mana%ers tr to ensure that the 8o is done ri%ht the

"irst time and that no ma8or a##idents o##ur on the *ro8e#t.

As ith #ost #ontrol9 the most im*ortant de#isions re%ardin% the =ualit o" a #om*leted

"a#ilit are made durin% the desi%n and *lannin% sta%es rather than durin% #onstru#tion. 7t is

durin% these *reliminar sta%es that #om*onent #on"i%urations9 material s*e#i"i#ations and

"un#tional *er"orman#e are de#ided. Qualit #ontrol durin% #onstru#tion #onsists lar%el o"

insurin% #on"orman#e to these ori%inal desi%n and *lannin% de#isions.

(hile #on"orman#e to e;istin% desi%n de#isions is the *rimar "o#us o" =ualit #ontrol9 there

are e;#e*tions to this rule. 4irst9 un"oreseen #ir#umstan#es9 in#orre#t desi%n de#isions or

#han%es desired an o ner in the "a#ilit "un#tion ma re=uire re5evaluation o" desi%n

de#isions durin% the #ourse o" #onstru#tion. (hile these #han%es ma e motivated the

#on#ern "or =ualit 9 the re*resent o##asions "or re5desi%n ith all the attendant o 8e#tives

and #onstraints. As a se#ond #ase9 some desi%ns rel u*on in"ormed and a**ro*riate de#ision

ma)in% durin% the #onstru#tion *ro#ess itsel". 4or e;am*le9 some tunnellin% methods ma)e

de#isions a out the amount o" shorin% re=uired at di""erent lo#ations ased u*on o servation

o" soil #onditions durin% the tunnellin% *ro#ess. !in#e su#h de#isions are ased on etter

in"ormation #on#ernin% a#tual site #onditions9 the "a#ilit desi%n ma e more #ost e""e#tive

as a result.

(ith the attention to #on"orman#e as the measure o" =ualit durin% the #onstru#tion *ro#ess9

the s*e#i"i#ation o" =ualit re=uirements in the desi%n and #ontra#t do#umentation e#omes

e;tremel im*ortant. Qualit re=uirements should e #lear and veri"ia le9 so that all *arties

in the *ro8e#t #an understand the re=uirements "or #on"orman#e.


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!a"et durin% the #onstru#tion *ro8e#t is also in"luen#ed in lar%e *art de#isions made

durin% the *lannin% and desi%n *ro#ess. !ome desi%ns or #onstru#tion *lans are inherentl

di""i#ult and dan%erous to im*lement9 hereas other9 #om*ara le *lans ma #onsidera l

redu#e the *ossi ilit o" a##idents. 4or e;am*le9 #lear se*aration o" tra""i# "rom #onstru#tion

&ones durin% road a reha ilitation #an %reatl redu#e the *ossi ilit o" a##idental

#ollisions. Be ond these desi%n de#isions9 sa"et lar%el de*ends u*on edu#ation9 vi%ilan#e

and #oo*eration durin% the #onstru#tion *ro#ess. (or)ers should e #onstantl alert to the

*ossi ilities o" a##idents and avoid ta)en unne#essar ris)s.

(.0 O&)a i*i ) "$& Quality a ! Sa"#ty

A variet o" di""erent or%ani&ations are *ossi le "or =ualit and sa"et #ontrol durin%

#onstru#tion. One #ommon model is to have a %rou* res*onsi le "or =ualit assuran#e and

another %rou* *rimaril res*onsi le "or sa"et ithin an or%ani&ation. 7n lar%e or%ani&ations9

de*artments dedi#ated to =ualit assuran#e and to sa"et mi%ht assi%n s*e#i"i# individuals to

assume res*onsi ilit "or these "un#tions on *arti#ular *ro8e#ts. 4or smaller *ro8e#ts9 the

*ro8e#t mana%er or an assistant mi%ht assume these and other res*onsi ilities. 7n either #ase9

insurin% sa"e and =ualit #onstru#tion is a #on#ern o" the *ro8e#t mana%er in overall #har%e o"

the *ro8e#t in addition to the #on#erns o" *ersonnel9 #ost9 time and other mana%ement issues.

7ns*e#tors and =ualit assuran#e *ersonnel ill e involved in a *ro8e#t to re*resent a variet

o" di""erent or%ani&ations. Ea#h o" the *arties dire#tl #on#erned ith the *ro8e#t ma have

their o n =ualit and sa"et ins*e#tors9 in#ludin% the o ner9 the en%ineer:ar#hite#t9 and the

various #onstru#tor "irms. ,hese ins*e#tors ma e #ontra#tors "rom s*e#iali&ed =ualit

assuran#e or%ani&ations. 7n addition to on5site ins*e#tions9 sam*les o" materials ill

#ommonl e tested s*e#iali&ed la oratories to insure #om*lian#e. 7ns*e#tors to insure

#om*lian#e ith re%ulator re=uirements ill also e involved. Common e;am*les areins*e#tors "or the lo#al %overnment?s uildin% de*artment9 "or environmental a%en#ies9 and

"or o##u*ational health and sa"et a%en#ies.

,he O##u*ational !a"et and @ealth Administration O!@A routinel #ondu#ts site visits o"

or) *la#es in #on8un#tion ith a**roved state ins*e#tion a%en#ies. O!@A ins*e#tors are

re=uired la to issue #itations "or all standard violations o served. !a"et standards

*res#ri e a variet o" me#hani#al sa"e%uards and *ro#edures "or e;am*le9 ladder sa"et is

#overed over 1'0 re%ulations. 7n #ases o" e;treme non5#om*lian#e ith standards9 O!@A


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ins*e#tors #an sto* or) on a *ro8e#t. @o ever9 onl a small "ra#tion o" #onstru#tion sites

are visited O!@A ins*e#tors and most #onstru#tion site a##idents are not #aused

violations o" e;istin% standards. As a result9 sa"et is lar%el the res*onsi ilit o" the

mana%ers on site rather than that o" *u li# ins*e#tors.

(hile the multitude o" *arti#i*ants involved in the #onstru#tion *ro#ess re=uire the servi#es

o" ins*e#tors9 it #annot e em*hasi&ed too stron%l that ins*e#tors are onl a "ormal #he#) on

=ualit #ontrol. Qualit #ontrol should e a *rimar o 8e#tive "or all the mem ers o" a *ro8e#t

team. Mana%ers should ta)e res*onsi ilit "or maintainin% and im*rovin% =ualit #ontrol.

Em*lo ee *arti#i*ation in =ualit #ontrol should e sou%ht and re arded9 in#ludin% the

introdu#tion o" ne ideas. Most im*ortant o" all9 =ualit im*rovement #an serve as a #atal st

"or im*roved *rodu#tivit . B su%%estin% ne or) methods9 avoidin% re or)9 and

avoidin% lon% term *ro lems9 %ood =ualit #ontrol #an *a "or itsel". O ners should *romote

%ood =ualit #ontrol and see) out #ontra#tors ho maintain su#h standards.

7n addition to the various or%ani&ational odies involved in =ualit #ontrol9 issues o" =ualit

#ontrol arise in virtuall all the "un#tional areas o" #onstru#tion a#tivities. 4or e;am*le9

insurin% a##urate and use"ul in"ormation is an im*ortant *art o" maintainin% =ualit

*er"orman#e. Other as*e#ts o" =ualit #ontrol in#lude do#ument #ontrol in#ludin% #han%es

durin% the #onstru#tion *ro#ess 9 *ro#urement9 "ield ins*e#tion and testin%9 and "inal

#he#)out o" the "a#ilit .

+.0 W$&, a ! Mat#&ial S-#%i"i%ati$ '

!*e#i"i#ations o" or) =ualit are an im*ortant "eature o" "a#ilit desi%ns. !*e#i"i#ations o"

re=uired =ualit and #om*onents re*resent *art o" the ne#essar do#umentation to des#ri e a

"a#ilit . , *i#all 9 this do#umentation in#ludes an s*e#ial *rovisions o" the "a#ilit desi%n as

ell as re"eren#es to %enerall a##e*ted s*e#i"i#ations to e used durin% #onstru#tion.

Constru#tion s*e#i"i#ations normall #onsist o" a series o" instru#tions or *rohi itions "or

s*e#i"i# o*erations. 4or e;am*le9 the "ollo in% *assa%e illustrates a t *i#al s*e#i"i#ation9 in

this #ase "or e;#avation "or stru#turesD

Con"orm to elevations and dimensions sho n on *lan ithin a toleran#e o" *lus or minus

0.10 "oot9 and e;tendin% a su""i#ient distan#e "rom "ootin%s and "oundations to *ermit *la#in%and removal o" #on#rete "orm or)9 installation o" servi#es9 other #onstru#tion9 and "or


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ins*e#tion. 7n e;#avatin% "or "ootin%s and "oundations9 ta)e #are not to distur ottom o"

e;#avation. E;#avate hand to "inal %rade 8ust e"ore #on#rete rein"or#ement is *la#ed.

,rim ottoms to re=uired lines and %rades to leave solid ase to re#eive #on#rete.

,his set o" s*e#i"i#ations re=uires 8ud%ment in a**li#ation sin#e some items are not *re#isel

s*e#i"ied. 4or e;am*le9 e;#avation must e;tend a su""i#ient distan#e to *ermit ins*e#tion

and other a#tivities. O viousl 9 the term su""i#ient in this #ase ma e su 8e#t to var in%

inter*retations. 7n #ontrast9 a s*e#i"i#ation that toleran#es are ithin *lus or minus a tenth o"

a "oot is su 8e#t to dire#t measurement. @o ever9 s*e#i"i# re=uirements o" the "a#ilit or

#hara#teristi#s o" the site ma ma)e the standard toleran#e o" a tenth o" a "oot ina**ro*riate.

(ritin% s*e#i"i#ations t *i#all re=uires a trade5o"" et een assumin% reasona le ehaviour

on the *art o" all the *arties #on#erned in inter*retin% ords su#h as su""i#ient versus the

e""ort and *ossi le ina##ura# in *re5s*e#i" in% all o*erations.

7n re#ent ears9 *er"orman#e s*e#i"i#ations have een develo*ed "or man #onstru#tion

o*erations. Rather than s*e#i" in% the re=uired #onstru#tion *ro#ess9 these s*e#i"i#ations

re"er to the re=uired *er"orman#e or =ualit o" the "inished "a#ilit . ,he e;a#t method

hi#h this *er"orman#e is o tained is le"t to the #onstru#tion #ontra#tor. 4or e;am*le9

traditional s*e#i"i#ations "or as*halt *avement s*e#i"ied the #om*osition o" the as*halt

material9 the as*halt tem*erature durin% *avin%9 and #om*a#tin% *ro#edures. 7n #ontrast9 a

*er"orman#e s*e#i"i#ation "or as*halt ould detail the desired *er"orman#e o" the *avement

ith res*e#t to im*ermea ilit 9 stren%th9 et#. @o the desired *er"orman#e level as attained

ould e u* to the *avin% #ontra#tor. 7n some #ases9 the *a ment "or as*halt *avin% mi%ht

in#rease ith etter =ualit o" as*halt e ond some minimum level o" *er"orman#e.

!xample "# Concrete $avement Strength

Con#rete *avements o" su*erior stren%th result in #ost savin%s dela in% the time at hi#h

re*airs or re5#onstru#tion is re=uired. 7n #ontrast9 #on#rete o" lo er =ualit ill ne#essitate

more "re=uent overla s or other re*air *ro#edures. Contra#t *rovisions ith ad8ustments to

the amount o" a #ontra#tor?s #om*ensation ased on *avement =ualit have e#ome

in#reasin%l #ommon in re#o%nition o" the #ost savin%s asso#iated ith hi%her =ualit

#onstru#tion. Even i" a *avement does not meet the ultimate desi%n standard9 it is still

orth usin% the lo er =ualit *avement and re5sur"a#in% later rather than #om*letel


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re8e#tin% the *avement. Based on these li"e # #le #ost #onsiderations9 a t *i#al *a s#hedule

mi%ht eDF1G

Table 1 - Load Ratio and Pay Factor

6oad Ratio Pa 4a#tor

H0.+0

0.+050. /

0.-050.3/

0./051.0/

1.1051.2/

1.$051.'/

I1.+0

Re8e#t

0./0

0./+

1.00

1.0+

1.10

1.12

7n this ta le9 the 6oad Ratio is the ratio o" the a#tual *avement stren%th to the desired desi%n

stren%th and the Pa 4a#tor is a "ra#tion hi#h the total *avement #ontra#t amount is

multi*lied to o tain the a**ro*riate #om*ensation to the #ontra#tor. 4or e;am*le9 i" a

#ontra#tor a#hieves #on#rete stren%th t ent *er#ent %reater than the desi%n s*e#i"i#ation9

then the load ratio is 1.20 and the a**ro*riate *a "a#tor is 1.0+9 so the #ontra#tor re#eives a

"ive *er#ent onus. 6oad "a#tors are #om*uted a"ter tests on the #on#rete a#tuall used in a

*avement. Note that a /0J *a "a#tor e;ists in this #ase ith even *avement =ualit onl

+0J o" that ori%inall desired. ,his hi%h *a "a#tor even ith ea) #on#rete stren%th mi%ht

e;ist sin#e mu#h o" the #ost o" *avements are in#urred in *re*arin% the *avement "oundation.

Con#rete stren%ths o" less then +0J are #ause "or #om*lete re8e#tion in this #ase9 ho ever.

.0 T$tal Quality C$ t&$l

Qualit #ontrol in #onstru#tion t *i#all involves insurin% #om*lian#e ith minimum

standards o" material and or)manshi* in order to insure the *er"orman#e o" the "a#ilit

a##ordin% to the desi%n. ,hese minimum standards are #ontained in the s*e#i"i#ations

des#ri ed in the *revious se#tion. 4or the *ur*ose o" insurin% #om*lian#e9 random sam*les

and statisti#al methods are #ommonl used as the asis "or a##e*tin% or re8e#tin% or)

#om*leted and at#hes o" materials. Re8e#tion o" a at#h is ased on non5#on"orman#e or
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violation o" the relevant desi%n s*e#i"i#ations. Pro#edures "or this =ualit #ontrol *ra#ti#e are

des#ri ed in the "ollo in% se#tions.

An im*li#it assum*tion in these traditional =ualit #ontrol *ra#ti#es is the notion o"

an a##e*ta le =ualit level hi#h is allo a le "ra#tion o" de"e#tive items. Materials o tained

"rom su**liers or or) *er"ormed an or%ani&ation is ins*e#ted and *assed as a##e*ta le i"

the estimated de"e#tive *er#enta%e is ithin the a##e*ta le =ualit level. Pro lems ith

materials or %oods are #orre#ted a"ter deliver o" the *rodu#t.

7n #ontrast to this traditional a**roa#h o" =ualit #ontrol is the %oal o" total =ualit #ontrol. 7n

this s stem9 no de"e#tive items are allo ed an here in the #onstru#tion *ro#ess. (hile the

&ero de"e#ts %oal #an never e *ermanentl o tained9 it *rovides a %oal so that anor%ani&ation is never satis"ied ith its =ualit #ontrol *ro%ram even i" de"e#ts are redu#ed

su stantial amounts ear a"ter ear. ,his #on#e*t and a**roa#h to =ualit #ontrol as "irst

develo*ed in manu"a#turin% "irms in a*an and Euro*e9 ut has sin#e s*read to man

#onstru#tion #om*anies. ,he est )no n "ormal #erti"i#ation "or =ualit im*rovement is the

7nternational Or%ani&ation "or !tandardi&ation?s 7!O /000 standard. 7!O /000 em*hasi&es

%ood do#umentation9 =ualit %oals and a series o" # #les o" *lannin%9 im*lementation and

revie .

,otal =ualit #ontrol is a #ommitment to =ualit e;*ressed in all *arts o" an or%ani&ation and

t *i#all involves man elements.


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ene"its that had een una**re#iated in traditional a**roa#hes. E;*enses asso#iated ith

inventor 9 re or)9 s#ra* and arranties ere redu#ed. (or)er enthusiasm and #ommitment

im*roved. Customers o"ten a**re#iated hi%her =ualit or) and ould *a a *remium "or

%ood =ualit . As a result9 im*roved =ualit #ontrol e#ame a #om*etitive advanta%e.

O" #ourse9 total =ualit #ontrol is di""i#ult to a**l 9 *arti#ular in #onstru#tion. ,he uni=ue

nature o" ea#h "a#ilit 9 the varia ilit in the or)"or#e9 the multitude o" su #ontra#tors and

the #ost o" ma)in% ne#essar investments in edu#ation and *ro#edures ma)e *ro%rams o"

total =ualit #ontrol in #onstru#tion di""i#ult. Nevertheless9 a #ommitment to im*roved

=ualit even ithout endorsin% the %oal o" &ero de"e#ts #an *a real dividends to

or%ani&ations.

!xample %# !xperience with Quality Circles

Qualit #ir#les re*resent a %rou* o" "ive to "i"teen or)ers ho meet on a "re=uent asis to

identi" 9 dis#uss and solve *rodu#tivit and =ualit *ro lems. A #ir#le leader a#ts as liaison

et een the or)ers in the %rou* and u**er levels o" mana%ement. A**earin% elo are

some e;am*les o" re*orted =ualit #ir#le a##om*lishments in #onstru#tionD F2G

On a hi%h a *ro8e#t under #onstru#tion ,aisei Cor*oration9 it as "ound that the lossrate o" read 5mi;ed #on#rete as too hi%h. A =ualit #ir#le #om*osed o" #ement masons

"ound out that the most im*ortant reason "or this as due to an ina##urate #he#)in% method.

B a**l in% the #ir#le?s re#ommendations9 the loss rate as redu#ed 11.'J.

7n a uildin% *ro8e#t !himi&u Constru#tion Com*an 9 ma #ases o" "ault rein"or#ed

#on#rete or) ere re*orted. ,he iron or)ers =ualit #ir#le e;amined their or)

thorou%hl and soon the "ault or)manshi* disa**eared. A 10J in#rease in *rodu#tivit

as also a#hieved.

/.0 Quality C$ t&$l y Stati'ti%al M#t $!'

An ideal =ualit #ontrol *ro%ram mi%ht test all materials and or) on a *arti#ular "a#ilit . 4or

e;am*le9 non5destru#tive te#hni=ues su#h as ;5ra ins*e#tion o" elds #an e used

throu%hout a "a#ilit . An on5site ins*e#tor #an itness the a**ro*riateness and ade=ua# o"

#onstru#tion methods at all times. Even etter9 individual #ra"tsmen #an *er"orm #ontinuin%

ins*e#tion o" materials and their o n or). E;haustive or 100J testin% o" all materials and
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or) ins*e#tors #an e e;#eedin%l e;*ensive9 ho ever. 7n man instan#es9 testin%

re=uires the destru#tion o" a material sam*le9 so e;haustive testin% is not even *ossi le. As a

result9 small sam*les are used to esta lish the asis o" a##e*tin% or re8e#tin% a *arti#ular

or) item or shi*ment o" materials. !tatisti#al methods are used to inter*ret the results o" test

on a small sam*le to rea#h a #on#lusion #on#ernin% the a##e*ta ilit o" an entire lot or at#h

o" materials or or) *rodu#ts.

,he use o" statisti#s is essential in inter*retin% the results o" testin% on a small sam*le.

(ithout ade=uate inter*retation9 small sam*le testin% results #an e =uite misleadin%. As an

e;am*le9 su**ose that there are ten de"e#tive *ie#es o" material in a lot o" one hundred. 7n

ta)in% a sam*le o" "ive *ie#es9 the ins*e#tor mi%ht not "ind an de"e#tive *ie#es or mi%ht

have all sam*le *ie#es de"e#tive.


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uildin% #om*onent is ina**ro*riate sin#e 8oints that are hard to rea#h ma e more li)el to

have ere#tion or "a ri#ation *ro lems.

Another assum*tion im*li#it in statisti#al =ualit #ontrol *ro#edures is that the =ualit o"

materials or or) is e;*e#ted to var "rom one *ie#e to another. ,his is #ertainl true in the

"ield o" #onstru#tion. (hile a desi%ner ma assume that all #on#rete is e;a#tl the same in a

uildin%9 the variations in material *ro*erties9 manu"a#turin%9 handlin%9 *ourin%9 and

tem*erature durin% settin% insure that #on#rete is a#tuall hetero%eneous in =ualit . Redu#in%

su#h variations to a minimum is one as*e#t o" =ualit #onstru#tion. 7nsurin% that the

materials a#tuall *la#ed a#hieve some minimum =ualit level ith res*e#t to avera%e

*ro*erties or "ra#tion o" de"e#tives is the tas) o" =ualit #ontrol.

2.0 Stati'ti%al Quality C$ t&$l 3it Sa4-li ) y Att&i ut#'

!am*lin% attri utes is a idel a**lied =ualit #ontrol method. ,he *ro#edure is intended

to determine hether or not a *arti#ular %rou* o" materials or or) *rodu#ts is a##e*ta le. 7n

the literature o" statisti#al =ualit #ontrol9 a %rou* o" materials or or) items to e tested is

#alled a lot or at#h. An assum*tion in the *ro#edure is that ea#h item in a at#h #an e tested

and #lassi"ied as either a##e*ta le or de"i#ient ased u*on mutuall a##e*ta le testin%

*ro#edures and a##e*tan#e #riteria. Ea#h lot is tested to determine i" it satis"ies a minimum

a##e*ta le =ualit level AQ6 e;*ressed as the ma;imum *er#enta%e o" de"e#tive items in a

lot or *ro#ess.

7n its asi# "orm9 sam*lin% attri utes is a**lied testin% a *re5de"ined num er o" sam*le

items "rom a lot. 7" the num er o" de"e#tive items is %reater than a tri%%er level9 then the lot is

re8e#ted as ein% li)el to e o" una##e*ta le =ualit . Other ise9 the lot is a##e*ted.


13/31

More "ormall 9 a lot is de"ined as a##e*ta le i" it #ontains a "ra#tion *1 or less de"e#tive

items. !imilarl 9 a lot is de"ined as una##e*ta le i" it #ontains a "ra#tion *2 or more de"e#tive

units. >enerall 9 the a##e*tan#e "ra#tion is less than or e=ual to the re8e#tion "ra#tion9 *1

*29 and the t o "ra#tions are o"ten e=ual so that there is no am i%uous ran%e o" lota##e*ta ilit et een *1 and *2. >iven a sam*le si&e and a tri%%er level "or lot re8e#tion or

a##e*tan#e9 e ould li)e to determine the *ro a ilities that a##e*ta le lots mi%ht e

in#orre#tl re8e#ted termed *rodu#er?s ris) or that de"i#ient lots mi%ht e in#orre#tl

a##e*ted termed #onsumer?s ris) .

Consider a lot o" "inite num er N9 in hi#h m items are de"e#tive ad and the remainin%

N5m items are non5de"e#tive %ood . 7" a random sam*le o" n items is ta)en "rom this lot9

then e #an determine the *ro a ilit o" havin% di""erent num ers o" de"e#tive items in the

sam*le. (ith a *re5de"ined a##e*ta le num er o" de"e#tive items9 e #an then develo* the

*ro a ilit o" a##e*tin% a lot as a "un#tion o" the sam*le si&e9 the allo a le num er o"

de"e#tive items9 and the a#tual "ra#tion o" de"e#tive items. ,his derivation a**ears elo .

,he num er o" di""erent sam*les o" si&e n that #an e sele#ted "rom a "inite *o*ulation N is

termed a mathemati#al #om ination and is #om*uted asD

1

here a "a#torial9 nL is n n51 n52 ... 1 and &ero "a#torial 0L is one #onvention. ,he

num er o" *ossi le sam*les ith e;a#tl ; de"e#tives is the #om ination asso#iated ith

o tainin% ; de"e#tives "rom m *ossi le de"e#tive items and n5; %ood items "rom N5m %ood

itemsD

2

>iven these *ossi le num ers o" sam*les9 the *ro a ilit o" havin% e;a#tl ; de"e#tive items

in the sam*le is %iven the ratio as the h *er%eometri# seriesD


14/31

$

(ith this "un#tion9 e #an #al#ulate the *ro a ilit o" o tainin% di""erent num ers o"

de"e#tives in a sam*le o" a %iven si&e.

!u**ose that the a#tual "ra#tion o" de"e#tives in the lot is * and the a#tual "ra#tion o" non5

de"e#tives is =9 then * *lus = is one9 resultin% in m N*9 and N 5 m N=. ,hen9 a "un#tion

% * re*resentin% the *ro a ilit o" havin% r or less de"e#tive items in a sam*le o" si&e n iso tained su stitutin% m and N into E=. $ and summin% over the a##e*ta le de"e#tive

num er o" itemsD

'

7" the num er o" items in the lot9 N9 is lar%e in #om*arison ith the sam*le si&e n9 then the

"un#tion % * #an e a**ro;imated the inomial distri utionD

+

or

,he "un#tion % * indi#ates the *ro a ilit o" a##e*tin% a lot9 %iven the sam*le si&e n and the

num er o" allo a le de"e#tive items in the sam*le r. ,he "un#tion % * #an e re*resented

%ra*hi#al "or ea#h #om ination o" sam*le si&e n and num er o" allo a le de"e#tive items r9as sho n in 4i%ure 1. Ea#h #urve is re"erred to as the o*eratin% #hara#teristi# #urve OC


15/31

#urve in this %ra*h. 4or the s*e#ial #ase o" a sin%le sam*le n 1 9 the "un#tion % * #an e

sim*li"iedD

1$.-

so that the *ro a ilit o" a##e*tin% a lot is e=ual to the "ra#tion o" a##e*ta le items in the lot.

4or e;am*le9 there is a *ro a ilit o" 0.+ that the lot ma e a##e*ted "rom a sin%le sam*le

test even i" "i"t *er#ent o" the lot is de"e#tive.


16/31

Figure 1- Example Operative Characteristic Curves ndicating Probability o! Lot "cceptance

4or an #om ination o" n and r9 e #an read o"" the value o" % * "or a %iven * "rom the

#orres*ondin% OC #urve. 4or e;am*le9 n 1+ is s*e#i"ied in 4i%ure 1. ,hen9 "or various

values o" r9 e "indD

r 0 * 2'J % * 2J


17/31

r 0

r 1

r 1

* 'J

* 2'J

* 'J

% * +'J

% * 10J

% * 33J

,he *rodu#er?s and #onsumer?s ris) #an e related to various *oints on an o*eratin%

#hara#teristi# #urve. Produ#er?s ris) is the #han#e that other ise acceptable lots "ail the

sam*lin% *lan i.e. have more than the allo a le num er o" de"e#tive items in the sam*le

solel due to random "lu#tuations in the sele#tion o" the sam*le. 7n #ontrast9 #onsumer?s ris)

is the #han#e that an una##e*ta le lot is a##e*ta le i.e. has less than the allo a le num er o"

de"e#tive items in the sam*le due to a etter than avera%e =ualit in the sam*le. 4or

e;am*le9 su**ose that a sam*le si&e o" 1+ is #hosen ith a tri%%er level "or re8e#tion o" one

item. (ith a "our *er#ent a##e*ta le level and a %reater than "our *er#ent de"e#tive "ra#tion9

the #onsumer?s ris) is at most ei%ht 5ei%ht *er#ent. 7n #ontrast9 ith a "our *er#ent a##e*ta le

level and a "our *er#ent de"e#tive "ra#tion9 the *rodu#er?s ris) is at most 1 5 0.33 0.12 or

t elve *er#ent.

7n s*e#i" in% the sam*lin% *lan im*li#it in the o*eratin% #hara#teristi# #urve9 the su**lier and

#onsumer o" materials or or) must a%ree on the levels o" ris) a##e*ta le to themselves. 7"

the lot is o" a##e*ta le =ualit 9 the su**lier ould li)e to minimi&e the #han#e or ris) that a

lot is re8e#ted solel on the asis o" a lo er than avera%e =ualit sam*le. !imilarl 9 the

#onsumer ould li)e to minimi&e the ris) o" a##e*tin% under the sam*lin% *lan a de"i#ient

lot. 7n addition9 oth *arties *resuma l ould li)e to minimi&e the #osts and dela s

asso#iated ith testin%.


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4or a t o *er#ent de"e#tive "ra#tion * 0.02 9 the resultin% a##e*tan#e value isD

sin% the inomial a**ro;imation in E=. + 9 the #om*ara le #al#ulation ould eD

hi#h is a di""eren#e o" 0.001/9 or 0.21 *er#ent "rom the a#tual value o" 0./020 "ound a ove.

7" the a##e*ta le de"e#tive *ro*ortion as t o *er#ent so * 1 * 2 0.02 9 then the #han#e o"

an in#orre#t re8e#tion or *rodu#er?s ris) is 1 5 % 0.02 1 5 0./ 0.1 or ten *er#ent. Note

that a *rudent *rodu#er should insure etter than minimum =ualit *rodu#ts to redu#e the

*ro a ilit or #han#e o" re8e#tion under this sam*lin% *lan. 7" the a#tual *ro*ortion o"

de"e#tives as one *er#ent9 then the *rodu#er?s ris) ould e onl "ive *er#ent ith this

sam*lin% *lan.

!xample # Designing a Sampling $lan

!u**ose that an o ner or *rodu#t #onsumer in the terminolo% o" =ualit #ontrol ishes

to have &ero de"e#tive items in a "a#ilit ith +9000 items o" a *arti#ular )ind. (hat ould e

the di""erent amounts o" #onsumer?s ris) "or di""erent sam*lin% *lans


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(ith an a##e*ta le =ualit level o" no de"e#tive items so * 1 0 9 the allo a le de"e#tive

items in the sam*le is &ero so r 0 in the sam*lin% *lan. sin% the inomial a**ro;imation9

the *ro a ilit o" a##e*tin% the +9000 items as a "un#tion o" the "ra#tion o" a#tual de"e#tive

items and the sam*le si&e isD

,o insure a ninet *er#ent #han#e o" re8e#tin% a lot ith an a#tual *er#enta%e de"e#tive o" one

*er#ent * 0.01 9 the re=uired sam*le si&e ould e #al#ulated asD

,hen9

As #an e seen9 lar%e sam*le si&es are re=uired to insure relativel lar%e *ro a ilities o" &ero

de"e#tive items.

5.0 Stati'ti%al Quality C$ t&$l 3it Sa4-li ) y 6a&ia l#'

As des#ri ed in the *revious se#tion9 sam*lin% attri utes is ased on a #lassi"i#ation o"

items as good or de!ective . Man or) and material attri utes *ossess #ontinuous *ro*erties9

su#h as stren%th9 densit or len%th. (ith the sam*lin% attri utes *ro#edure9 a *arti#ular

level o" a varia le =uantit must e de"ined as a##e*ta le =ualit . More %enerall 9 t o items

#lassi"ied as good mi%ht have =uite di""erent stren%ths or other attri utes. 7ntuitivel 9 it seemsreasona le that some #redit should e *rovided "or e;#e*tionall %ood items in a sam*le.

!am*lin% varia les as develo*ed "or a**li#ation to #ontinuousl measura le =uantities

o" this t *e. ,he *ro#edure uses measured values o" an attri ute in a sam*le to determine the

overall a##e*ta ilit o" a at#h or lot. !am*lin% varia les has the advanta%e o" usin% more

in"ormation "rom tests sin#e it is ased on a#tual measured values rather than a sim*le

#lassi"i#ation. As a result9 a##e*tan#e sam*lin% varia les #an e more e""i#ient than

sam*lin% attri utes in the sense that "e er sam*les are re=uired to o tain a desired levelo" =ualit #ontrol.


20/31

7n a**l in% sam*lin% varia les9 an a##e*ta le lot =ualit #an e de"ined ith res*e#t to an

u**er limit 9 a lo er limit 69 or oth. (ith these oundar #onditions9 an a##e*ta le =ualit

level #an e de"ined as a ma;imum allo a le "ra#tion o" de"e#tive items9 M. 7n 4i%ure 29 the

*ro a ilit distri ution o" item attri ute ; is illustrated. (ith an u**er limit 9 the "ra#tion o"

de"e#tive items is e=ual to the area under the distri ution "un#tion to the ri%ht o" so that

; . ,his "ra#tion o" de"e#tive items ould e #om*ared to the allo a le "ra#tion M to

determine the a##e*ta ilit o" a lot. (ith oth a lo er and an u**er limit on a##e*ta le

=ualit 9 the "ra#tion de"e#tive ould e the "ra#tion o" items %reater than the u**er limit or

less than the lo er limit. Alternativel 9 the limits #ould e im*osed u*on the

a##e*ta le average level o" the varia le


21/31

Figure # - $ariable Probability %istributions and "cceptance Regions

7n sam*lin% varia les9 the "ra#tion o" de"e#tive items is estimated usin% measured

values "rom a sam*le o" items. As ith sam*lin% attri utes9 the *ro#edure assumes a

random sam*le o" a %ive si&e is o tained "rom a lot or at#h. 7n the a**li#ation o" sam*lin%

varia les *lans9 the measured #hara#teristi# is virtuall al a s assumed to e normally

distributed as illustrated in 4i%ure 2. ,he normal distri ution is li)el to e a reasona l %ood

assum*tion "or man measured #hara#teristi#s su#h as material densit or de%ree o" soil

#om*a#tion. ,he Central 6imit ,heorem *rovides a %eneral su**ort "or the assum*tionD i" the

sour#e o" variations is a lar%e num er o" small and inde*endent random e""e#ts9 then the


22/31

resultin% distri ution o" values ill a**ro;imate the normal distri ution. 7" the distri ution o"

measured values is not li)el to e a**ro;imatel normal9 then sam*lin% attri utes should

e ado*ted.


23/31

the de%rees o" "reedom *arameter increases . As the num er o" de%rees o" "reedom e#omes

ver lar%e9 the t5distri ution #oin#ides ith the normal distri ution.

(ith an u**er limit9 the #al#ulations are similar9 and the *ro a ilit that the avera%e value o"

a *o*ulation is less than a *arti#ular u**er limit #an e #al#ulated "rom the test statisti#D

11

(ith oth u**er and lo er limits9 the sum o" the *ro a ilities o" ein% a ove the u**er limit

or elo the lo er limit #an e #al#ulated.

,he #al#ulations to estimate the "ra#tion o" items a ove an u**er limit or elo a lo er limit

are ver similar to those "or the *o*ulation avera%e. ,he onl di""eren#e is that the s=uare

root o" the num er o" sam*les does not a**ear in the test statisti# "ormulasD

12

and

1$

here t A6 is the test statisti# "or all items ith a lo er limit and t A is the test statisti# "or allitems ith a u**er limit. 4or e;am*le9 the test statisti# "or items a ove an u**er limit o" +.+

ith '.09 s $.09 and n + is t A 3.+ 5 '.0 :$.0 1.+ ith n 5 1 ' de%rees o"

"reedom.

7nstead o" usin% sam*lin% *lans that s*e#i" an allo a le "ra#tion o" de"e#tive items9 it saves

#om*utations to sim*l rite s*e#i"i#ations in terms o" the allo a le test statisti# values

themselves. ,his *ro#edure is e=uivalent to re=uirin% that the sam*le avera%e e at least a

*re5s*e#i"ied num er o" standard deviations a a "rom an u**er or lo er limit. 4or e;am*le9


24/31

ith '.09 3.+9 s $.0 and n '19 the sam*le mean is onl a out 3.+ 5 '.0 :$.0 1.+

standard deviations a a "rom the u**er limit.

,o summari&e9 the a**li#ation o" sam*lin% varia les re=uires the s*e#i"i#ation o" a sam*le

si&e9 the relevant u**er or limits9 and either 1 the allo a le "ra#tion o" items "allin% outside

the desi%nated limits or 2 the allo a le *ro a ilit that the *o*ulation avera%e "alls outside

the desi%nated limit. Random sam*les are dra n "rom a *re5de"ined *o*ulation and tested to

o tained measured values o" a varia le attri ute. 4rom these measurements9 the sam*le

mean9 standard deviation9 and =ualit #ontrol test statisti# are #al#ulated. 4inall 9 the test

statisti# is #om*ared to the allo a le tri%%er level and the lot is either a##e*ted or re8e#ted. 7t

is also *ossi le to a**l se=uential sam*lin% in this *ro#edure9 so that a at#h ma e

su 8e#ted to additional sam*lin% and testin% to "urther re"ine the test statisti# values.

(ith sam*lin% varia les9 it is nota le that a *rodu#er o" material or or) #an ado*t t o

%eneral strate%ies "or meetin% the re=uired s*e#i"i#ations. 4irst9 a *rodu#er ma insure that

the avera%e =ualit level is =uite hi%h9 even i" the varia ilit amon% items is hi%h. ,his

strate% is illustrated in 4i%ure $ as a hi%h =ualit avera%e strate% . !e#ond9 a *rodu#er

ma meet a desired =ualit tar%et redu#in% the variability ithin ea#h at#h. 7n 4i%ure $9

this is la eled the lo varia ilit strate% . 7n either #ase9 a *rodu#er should maintain hi%hstandards to avoid re8e#tion o" a at#h.


25/31

Figure (- Testing !or %e!ective Component )trengths

!xample # *esting for defective component strengths

!u**ose that an ins*e#tor ta)es ei%ht stren%th measurements ith the "ollo in% resultsD

'.$9 '.39 '. 9 '.-9 '.'9 '. 9 '.-9 '.


26/31


27/31

Table #- +on!atal Occupational n*ury and llness ncidence Rates

7ndustr 1// 200

A%ri#ulture9 "orestr 9 "ishin%

Minin%Constru#tion

Manu"a#turin%

,rade9,rans*ortation and utilities

4inan#ial a#tivities

Pro"essional and usiness servi#es

3.-

+.'/./

10.

3.-

2.'

.0

$.++./

+

1.+

1.2

NoteD


28/31

Table (- Fatal Occupational n*uries in Construction 1../ and #00

Year 1//- 200'

,otal "atalities

4alls,rans*ortation in#idents

Conta#t ith o 8e#ts S

e=ui*ment

E;*osure to harm"ul su stan#es

and environments

1910-$-

233

1//

133

192$'''+

23-

2 -

1-0

!our#eD Bureau o" 6a or !tatisti#s

Table - Fatality Causes in Construction 1..231../ and #0023#00/

Year / :/- 0 :0-

,otal a##idents

4alls "rom a hei%ht

!tru#) a movin% vehi#le

!tru#) movin%:"allin% o 8e#t

,ra**ed somethin% overturnin%:#olla*sin%


29/31

e;*anded "or #onstru#tion e=ui*ment and tools. Materials and or) *ro#ess #hoi#es also

in"luen#e the sa"et o" #onstru#tion. 4or e;am*le9 su stitution o" alternative materials "or

as estos #an redu#e or eliminate the *ros*e#ts o" lon% term illnesses su#h as asbestiosis .

Edu#atin% or)ers and mana%ers in *ro*er *ro#edures and ha&ards #an have a dire#t im*a#t

on 8o site sa"et . ,he reali&ation o" the lar%e #osts involved in #onstru#tion in8uries and

illnesses *rovides a #onsidera le motivation "or a areness and edu#ation. Re%ular sa"et

ins*e#tions and sa"et meetin%s have e#ome standard *ra#ti#es on most 8o sites.

Pre5=uali"i#ation o" #ontra#tors and su 5#ontra#tors ith re%ard to sa"et is another im*ortant

avenue "or sa"et im*rovement. 7" #ontra#tors are onl invited to id or enter ne%otiations i"

the have an a##e*ta le re#ord o" sa"et as ell as =ualit *er"orman#e 9 then a dire#tin#entive is *rovided to insure ade=uate sa"et on the *art o" #ontra#tors.


30/31

*e#uliarities and as a result o" e;a#tl these s*e#ial *ro lems9 im*rovin% or)site sa"et is a

ver im*ortant *ro8e#t mana%ement #on#ern.

8.0 R#"# %#'

1. An%9 A.@.!. and (.@. ,an%9 Probability Concepts in Engineering Planning and

%esign4 $olume - 5asic Principles 9 ohn (ile and !ons9 7n#.9 Ne Yor)9 1/-+.

2. Au9 ,.9 R.M. !hane9 and 6.A. @oel9 Fundamentals o! )ystems Engineering4

Probabilistic 6odels 9 Addison5(esle Pu lishin% Co.9 Readin% MA9 1/-2

$. Bo )er9 A.@. and 6ie ermann9 >. .9 Engineering )tatistics 9 Prenti#e5@all9 1/-2.

'. 4o;9 A. . and Cornell9 @.A.9 eds 97uality in the Constructed Pro*ect Ameri#an

!o#iet o" Civil En%ineers9 Ne Yor)9 1/3'.

+. 7nternational Or%ani&ation "or !tandardi&ation9 !am*lin% Pro#edures and Charts "or

7ns*e#tion aria les "or Per#ent


31/31

10.0 9$$t $t#'

1. ,his illustrative *a "a#tor s#hedule is ada*ted "rom R.M. (eed9 ill 9 A. ,ouran9 and ,. Asai9 Qualit Control Cir#les in Constru#tion9 ")CE

;ournal o! Construction Engineering and 6anagement 9 ol. 11$9 No. $9 1/3-9 *% '$2.

$. !ee mproving Construction )a!ety Per!ormance 9 Re*ort A5$9 ,he Business Roundta le9

Ne Yor)9 NY9 anuar 1/32.

'. @in&e9 immie (.9 Construction )a!ety 9 Prenti#e5@all9 1//-.

13 Eng Maths Editied

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