Levine Smume6 Ppt06 Shortened

8/16/2019 Levine Smume6 Ppt06 Shortened

http://slidepdf.com/reader/full/levine-smume6-ppt06-shortened 1/45

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-1

Chapter 6

he !or"al #istribution $ %therContinuous #istributions

Statistics for Managers usingMicrosoft Excel 6 th Edition




&earning %b'ecti(es

In this chapter, you learn: o co"pute probabilities )ro" the nor"al distribution

Ho* to use the nor"al distribution to sol(e business

proble"so use the nor"al probability plot to deter"ine *hether

a set o) data is appro+i"ately nor"ally distributed



Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-

Continuous Probability #istributions

continuous rando" (ariable is a (ariable thatcan assu"e any (alue on a continuu" canassu"e an uncountable nu"ber o) (alues/

thic ness o) an ite"ti"e re uired to co"plete a taste"perature o) a solutionheight, in inches

hese can potentially ta e on any (aluedepending only on the ability to precisely andaccurately "easure




he !or"al #istribution

‘Bell Shaped ’ Symmetrical Mean, Median and Mode

are EqualLocation is determined by themean, Spread is determined by thestandard de!iation, "

#he random !ariable has anin$inite theoretical ran%e:3 ∞ to ∞

Mean& Median& Mode

'

$(')

4




he !or"al #istribution#ensity unction

he )or"ula )or the nor"al probability density )unction is

Where e = the mathematical constant approximated by 2.71828π = the mathematical constant approximated by 3.14159μ = the pop lation mean! = the pop lation standard de"iation# = any "al e o$ the contin o s "ariable

2μ%&#21

e2π1$&#%

−

−= σ

σ




By !aryin% the parameters and " , *e obtaindi$$erent normal distributions

7any !or"al #istributions


http://slidepdf.com/reader/full/levine-smume6-ppt06-shortened 7/45Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-8

he !or"al #istribution9hape

'

$(')

"

Changing shi)ts thedistribution le)t or right .

Changing 4 increasesor decreases thespread.


http://slidepdf.com/reader/full/levine-smume6-ppt06-shortened 8/45Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-:

he 9tandardi;ed !or"al

ny nor"al distribution *ith any "ean andstandard de(iation co"bination/ can betrans)or"ed into the standardi;ed nor"al distribution </

!eed to trans)or" = units into < units

he standardi;ed nor"al distribution </ has a"ean o) 0 and a standard de(iation o) 1


http://slidepdf.com/reader/full/levine-smume6-ppt06-shortened 9/45Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6->

ranslation to the 9tandardi;ed!or"al #istribution

ranslate )ro" = to the standardi;ed nor"althe ?<@ distribution/ bysubtracting the "ean

o) = and di(iding by its standard de(iation A

he < distribution al*ays has "ean B 0 and

standard de(iation B 1

4

C=< −=



he 9tandardi;ed !or"alProbability #ensity unction

he )or"ula )or the standardi;ed nor"alprobability density )unction is

Dhere e B the "athe"atical constant appro+i"ated by 2.81:2: B the "athe"atical constant appro+i"ated by .1 15>

< B any (alue o) the standardi;ed nor"al distribution

21F2/<e2E1) </ −=



he 9tandardi;ed !or"al #istribution

lso no*n as the ?<@ distribution7ean is 0

9tandard #e(iation is 1

+

$(+)

-

Galues abo(e the "ean ha(e positi(e <-(alues,(alues belo* the "ean ha(e negati(e <-(alues



E+a"ple

I) = is distributed nor"ally *ith "ean o) 100 and standard de(iation o) 50 , the < (alue)or = B 200 is

his says that = B 200 is t*o standardde(iations 2 incre"ents o) 50 units/ abo(ethe "ean o) 100.

2.'(5'

1''((2''!

μ#) =

−=

−=



Co"paring = and < units

+.-

/0./ .'

1ote that the shape o$ the distribution is the same,only the scale has chan%ed0 2e can e3press theproblem in the ori%inal units (' in dollars) or in

standardi4ed units (+)

B 100, 4 B 50/

B 0, 4 B 1/



inding !or"al Probabilities

a b '

$(') P a ' b /5

Probability is "easured by the areaunder the cur(e

5

P a ' b /B!ote that the probability

o) any indi(idual (alue is;ero/



$(')

'

Probability as rea nder the Cur(e

0.50.5

he total area under the cur(e is 1.0 , and the cur(e issy""etric, so hal) is abo(e the "ean, hal) is belo*

1.0/=P =∞<<−∞

0.5/=P C =∞<<0.5C/=P =<<−∞



he 9tandardi;ed !or"al able

he Cu"ulati(e 9tandardi;ed !or"al tablein the te+tboo ppendi+ table E.2/ gi(es theprobability less than a desired (alue o) < i.e.,)ro" negati(e in)inity to </

+/0

0788/E+a"pleA

P < J 2.00/ B 0.>882



he 9tandardi;ed !or"al able

he (alue *ithin the

table gi(es theprobability )ro" < B ∞ up to the desired <(alue

0788/

2.09(+ /0 ) & 0788/

he ro* sho*sthe (alue o) <to the )irst

deci"al point

he colu"n gi(es the (alue o)< to the second deci"al point

/0

..

.

(continued)

< 0.00 0.01 0.02 K

0.0

0.1


http://slidepdf.com/reader/full/levine-smume6-ppt06-shortened 18/45Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-1:

Leneral Procedure )orinding !or"al Probabilities

#ra* the nor"al cur(e )or the proble" in ter"s o) =

ranslate =-(alues to <-(alues

se the 9tandardi;ed !or"al able

o )ind P a J = J b/ *hen = isdistributed nor"allyA


http://slidepdf.com/reader/full/levine-smume6-ppt06-shortened 19/45Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-1>

inding !or"al Probabilities

&et = represent the ti"e it ta es in seconds/to do*nload an i"age )ile )ro" the internet.9uppose = is nor"al *ith a "ean o)1:.0seconds and a standard de(iation o) 5.0seconds. ind P = J 1:.6/

- 06

'- 0



&et = represent the ti"e it ta es, in seconds to do*nload an i"age )ile)ro" the internet.9uppose = is nor"al *ith a "ean o) 1:.0 seconds and a standardde(iation o) 5.0 seconds. ind P = J 1:.6/

+0-/'- 06-

B 1: 4 B 5

& " & -

(continued)inding !or"al Probabilities

P = J 1:.6/ P < J 0.12/

'.125.'8.'118.*

!μ#

) =

−=

−=



+

0-/

< .00 .01

0.0 .5000 .50 0 .50:0

.5 >: .5 :

0.2 .58> .5: 2 .5:81

0. .618> .6218 .6255

9olutionA inding P < J 0.12/

0.5 8:0 /

0- 05 8:

9tandardi;ed !or"al Probabilityable Portion/

0

B P < J 0.12/P = J 1:.6/



inding !or"alpper ail Probabilities

9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.

!o* ind P = M 1:.6/

'

- 06

- 0



!o* ind P = M 1:.6/K(continued)

+

0-/

+

0-/

0.5 8:

1.000 1.0 - 0.5 8:B 0. 522

P = M 1:.6/ B P < M 0.12/ B 1.0 - P < N 0.12/

B 1.0 - 0.5 8: B 0. 522

inding !or"alpper ail Probabilities




inding a !or"al ProbabilityOet*een *o Galues

9uppose = is nor"al *ith "ean 1:.0 andstandard de(iation 5.0. ind P 1: J = J 1:.6/

P 1: J = J 1:.6/

B P 0 J < J 0.12/

+0-/'- 06-

Calculate <-(aluesA

'5

8118!

μ#) =−=−=

'.125

8118.*!

μ#) =−=−=




+

0-/

9olutionA inding P 0 J < J 0.12/

0.0 8:

0

B P 0 J < J 0.12/P 1: J = J 1:.6/

B P < J 0.12/ P < N 0/

B 0.5 8: - 0.5000 B 0.0 8:

0.5000

< .00 .01

0.0 .5000 .50 0 .50:0

.5 >: .5 :

0.2 .58> .5: 2 .5:81

0. .618> .6218 .6255

0 /

0- 05 8:





9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.

!o* ind P 18. J = J 1:/

'

-80;- 0

Probabilities in the &o*er ail




Probabilities in the &o*er ail

!o* ind P 18. J = J 1:/K

'-80; - 0

P 18. J = J 1:/

B P -0.12 J < J 0/B P < J 0/ P < N -0.12/

B 0.5000 - 0. 522 B 0.0 8:

(continued)

0.0 8:

0. 522

+< 0-/

he !or"al distribution issy""etric, so this probabilityis the sa"e as P 0 J < J 0.12/



Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-2:

E"pirical Qules

= -" encloses about6 0/6> o$ '’s

$(')

'?-"<-"

Dhat can *e say about the distribution o) (aluesaround the "eanR or any nor"al distributionA

""

6 0/6>



Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-2>

he E"pirical Qule

= /" co!ers about 7@> o$ '’s

= A" co!ers about 7708> o$ '’s

3

/" /"

3

A" A"

7@0;;> 7708A>

(continued)

( l b b l



Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6- 0

9teps to )ind the = (alue )or a no*nprobabilityA

1. ind the < (alue )or the no*n probability2. Con(ert to = units using the )or"ulaA

Li(en a !or"al Probabilityind the = Galue

<4C= +=

i di h ( l )




inding the = (alue )or aSno*n Probability

E+a"pleA&et = represent the ti"e it ta es in seconds/ todo*nload an i"age )ile )ro" the internet.

9uppose = is nor"al *ith "ean 1:.0 and standardde(iation 5.0ind = such that 20T o) do*nload ti"es are less than

=.

'- 0

0.2000

+

(continued)

i d h ( l )




ind the < (alue )or20T in the &o*er ail

20T area in the lo*ertail is consistent *ith a< (alue o) -0.:< .0

-0.> .1862 .18 6

.20-0.8 .2 28 .22>6

0 ;

< 0 02005


.05

.1811

.1>88

.2266

K

K

KK

'- 0

0.2000

+< 0 ;

1. ind the < (alue )or the no*n probability




2. Con(ert to = units using the )or"ulaA

inding the = (alue

9o 20T o) the (alues )ro" a distribution*ith "ean 1:.0 and standard de(iation5.0 are less than 1 .:0

8.13

'.5%84.'&'.18

)!μ#

=−+=

+=




E(aluating !or"ality

!ot all continuous distributions are nor"alIt is i"portant to e(aluate ho* *ell the data set isappro+i"ated by a nor"al distribution.!or"ally distributed data should appro+i"ate thetheoretical nor"al distributionA

he nor"al distribution is bell shaped sy""etrical/*here the "ean is e ual to the "edian.

he e"pirical rule applies to the nor"al distribution.

he inter uartile range o) a nor"al distribution is 1.standard de(iations.





Co"paring data characteristics to theoreticalproperties

Construct charts or graphsor s"all- or "oderate-si;ed data sets, construct a ste"-and-lea)

display or a bo+plot to chec )or sy""etryor large data sets, does the histogra" or polygon appear bell-

shapedR

Co"pute descripti(e su""ary "easures#o the "ean, "edian and "ode ha(e si"ilar (aluesRIs the inter uartile range appro+i"ately 1. 4RIs the range appro+i"ately 6 4R

(continued)





Co"paring data characteristics to theoreticalproperties %bser(e the distribution o) the data set

#o appro+i"ately 2F o) the obser(ations lie *ithin "ean U1standard de(iationR#o appro+i"ately :0T o) the obser(ations lie *ithin "eanU1.2: standard de(iationsR#o appro+i"ately >5T o) the obser(ations lie *ithin "ean U2standard de(iationsR

E(aluate nor"al probability plotIs the nor"al probability plot appro+i"ately linear i.e. a straightline/ *ith positi(e slopeR

(continued)

C i




Constructing !or"al Probability Plot

!or"al probability plot rrange data into ordered array

ind corresponding standardi;ed nor"al uantile(alues </

Plot the pairs o) points *ith obser(ed data (alues =/on the (ertical a+is and the standardi;ed nor"al

uantile (alues </ on the hori;ontal a+isE(aluate the plot )or e(idence o) linearity

h ! " l P b bilit Pl t



Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6- :

nor"al probability plot )or data)ro" a nor"al distribution *ill be

appro+i"ately linear A

0

60

>0

-2 -1 0 1 2 <

=

he !or"al Probability PlotInterpretation

! " l P b bilit Pl t



Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6- >

!or"al Probability PlotInterpretation

&e)t-9 e*ed Qight-9 e*ed

Qectangular

0

60

>0

-2 -1 0 1 2 <

=

(continued)

0

60

>0

-2 -1 0 1 2 <

=

0

60

>0

-2 -1 0 1 2 <

= !onlinear plots indicatea de(iation )ro"nor"ality




E(aluating !or"ality n E+a"pleA Oond unds Qeturns

he bo+plot is s e*ed tothe le)t. he nor"aldistribution is sy""etric./

- 0 - 0 -20 -10 0 10

Bond unds / Deturns

Bond unds / Deturns

i!e<number Summary

Minimum - 1.>irst uartile -2.5

Median 2.6

#hird uartile 8.1Ma3imum 15.0





#escripti(e 9tatistics

(continued)

• he "ean 1. 18/ is less than the "edian 2.6/.In a nor"al distribution the "ean and "edian

are e ual./• he inter uartile range o) >.6 is appro+i"ately

1.25 standard de(iations. In a nor"al

distribution the inter uartile range is 1.standard de(iations./

• he range o) 6.> is e ual to 6.1 standardde(iations. In a nor"al distribution the range is6 standard de(iations./

• 86.1T o) the obser(ations are *ithin 1 standardde(iation o) the "ean. In a nor"al distributionthis percentage is 6:.26T.

• ::. T o) the obser(ations are *ithin 1.2:standard de(iations o) the "ean. In a nor"aldistribution this percentage is :0T./





(continued)

Plot is not a straightline and sho*s the

distribution is s e*edto the le)t. henor"al distributionappears as a straightline./

( l l




E(aluating !or"ality n E+a"pleA 7utual unds Qeturns

Conclusionshe returns are le)t-s e*edhe returns ha(e "ore (alues concentrated around

the "ean than e+pectedhe range is larger than e+pected

!or"al probability plot is not a straight line%(erall, this data set greatly di))ers )ro" thetheoretical properties o) the nor"al distribution

(continued)




Chapter 9u""ary

Presented ey continuous distributions nor"al, uni)or", e+ponential

ound probabilities using )or"ulas and tables

Qecogni;ed *hen to apply di))erent distributions

pplied distributions to decision proble"s



ll rights reser(ed. !o part o) this publication "ay be reproduced, stored in a retrie(alsyste", or trans"itted, in any )or" or by any "eans, electronic, "echanical, photocopying,

recording, or other*ise, *ithout the prior *ritten per"ission o) the publisher.Printed in the nited 9tates o) "erica.

Levine Smume6 Ppt06 Shortened

Documents

Transcript of Levine Smume6 Ppt06 Shortened