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CENTRAL LIMIT THEOREM (CLT) - MEAN OF MEANS IS NORMAL (FIGURE 16.28)

# CENTRAL LIMIT THEOREM (CLT) - MEAN OF MEANS IS NORMAL (FIGURE 16.28)

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populationmeanμ.

Theclustersizencanbequitesmall,andthe

histogramofclustermeanvalues,Xm,willrapidly

convergetoanormaldistributionregardlessofthe

underlyingpopulation.

TheCentralLimitTheoremappliestoanypopulation

distribution,includingthediscreteandcontinuous

distributionsaswellasbimodaldistributions.

Whendiscretesamplingisinvolved,thedistributionof

averages(i.e.,themeanofclusters)mustbeused.

Variancegetssmallerasnincreases;thesmallerthe

numberofsamplesinaclusterthelargerthevariance

ofthemeans.

NORMALIZEDTRANSFORMS

1. Engineeringexperimentsoftenproduceresponsesthatmaynot

appearnormal,however,manyoftheseresponsescanoftenbe

transformedintonormallydistributedrepresentations.

2. Thenormaldistributionisemphasizedbecausemany

techniquesinstatisticsarebasedontheassumptionofa

normaldistribution.TwoverytypicaltransformationsareY=In

XandY= .However,therearemanymore.

3. Itisimportanttorecognizethatmanyseeminglynonnormal

distributionscanbemathematicallytransformedintowhat

appearstobeanormaldistribution.

4. Ausefulnormalizationisbasedontheassumptionofalarge

numberofbasicmeasurementsorobservationsaswellasthe

useoftheCentralLimitTheorem.

DISCRETEPROBABILITYDISTRIBUTIONS

BINOMIALDISTRIBUTION(BERNOULLI)

ThebinomialandPoissondistributionsarethemostcommon

distributionswithmanyapplications.KeycharacteristicsoftheBinomial

andPoissondistributionsare:

Binomialdistribution

Frequentlyusedinengineering

Probabilityofsuccesspandfailureq

Combinationsofp'sandq's

PoissonDistribution

Raresuccesses,pverysmall

Largesamplesize

Limitofbinomial

PopulationParameters:characteristicsofthepopulation

Populationsize:N

Probabilityofsuccess:p

Probabilityoffailure:q=1-p

Mean:μ

Variance:σ2

SampleStatistics:characteristicsofthesample

Randomvariable:Xi

Samplesize:n

Mean:

Variance:s2

Samplestakenwithoutreplacementaregenerallynotindependentsince

pisnotconstant.Ifsamplesizen<0.05Npopulationsize,wecan

considerpunchangedand"independent."

Binomialdistribution—ProbabilitywhenEXACTLYxoutofnevents

occur:

Theassumptionsare:

1. Experimentofnindependentevents.

2. Probabilityofa"success"isp.

3. Probabilitypisconstantforallevents.

4. Probabilityof"failure"isq=1-p.

5. Parametersnandparespecified.

6. Randomvariable×numberofsuccesses.

8. Orderofsuccessnotimportant;combination.

Mean:

Variance:

Example1

Sixtossesofacoin.

Experimentissixtossesofacoin,n=6.

Probabilityofatailinonetoss,q=(1-p)=1/2.

Randomvariable:x=0,1,2,3,4,5,6

C(x;n)

pxqn-x

0

(6!)/(0!6!)=1

(1/2)0(1/2)6=1/64

1/64

1

(6!)/(1!5!)=6

(1/2)1(1/2)5=1/64

6/64

2

(6!)/(2!4!)=15

(1/2)2(1/2)4=1/64

15/64

3

(6!)/(3!3!)=20

(1/2)3(1/2)3=1/64

20/64

4

(6!)/(4!2!)=15

(1/2)4(1/2)2=1/64

15/64

r.v.x.

B(x;n,p)

5

(6!)/(5!1!)=6

(1/2)5(1/2)1=1/64

6/64

6

(6!)/(6!0!)=1

(1/2)6(1/2)0=1/64

1/64

Sum=64/64

Ifweweretographthisdata,wewouldobtainthegraphshowninFigure

16.29.

Figure16.29:Binomialdistributionhistogram—sixtossesofa

coin.

Parameters:n=6;p=1/2;q=1/2

Mean:μ=np=6·1/2=3

Variance:σ2=npq=6·1/2·1/2=3/2=1.5

Notethatbecausep=1/2,(1)themeanisequaltothemid-range,and

Example2

Squarerodwithfoursides.

Sidesaredenoted1,2,3,4respectively.

Experimentissixtossesoffour-sidedrod,n=6.

Randomvariableistossingthenumber3.

Probabilityofa3forasingletoss,p=1/4.

Probabilityof"no3"inasingletoss,q=(1-p)=3/4.

Find:Probabilityofexactlytwo3sin6tosses;x=2.

Parameters:n=6tosses;successa"3,"hasp=1/4.

Randomvariable:x=0,1,2,3,4,5,6

C(x;n)

pxqn-x

(6!)/(0!6!)=1

(1/4)0(3/4)6=36/46

1

r.v.x.

B(x;n,p)

%

729/46

17.8

(6!)/(1!5!)=6

(1/4)1(3/4)5=

35/46

1458/46

35.6

2

(6!)/(2!4!)=15

(1/4)2(3/4)4=

34/46

1215/46

29.7

3

(6!)/(3!3!)=20

(1/4)3(3/4)3=

33/46

540/46

13.2

(6!)/(4!2!)=15

(1/4)4(3/4)2=

32/46

135/46

3.30

18/46

0.44

0

4

5

(6!)/(5!1!)=6

(1/4)5(3/4)1=

31/46

6

(6!)/(6!0!)=1

(1/4)6(3/4)0=1/46

Where46=

4096

1/46

0.02

SUM=4096/46

100

ThehistogramforthisdataisshowninpercentofB(x;n,p)inFigure

16.30.

Figure16.30:HistograminpercentofB(x;n,p).

Parameters:n=6,p=1/4,q=3/4

Mean:μ=np=6·1/4=3/2=1.5

Variance:σ2=npq=6·1/4·3/4=18/16=1.125

Example3

Squarerodwithfoursides.

Sidesaredenoted1,2,3,4,respectively.

Experimentissixtossesoffour-sidedrod,n=6.

Randomvariableistossinganynumberexcept3.

Probabilityof"no3"inasingletoss,p=3/4.

Probabilityofa3forasingletoss,q=(1-p)=1/4.

ThedataareshowngraphicallyinFigure16.31.

Figure16.31:Binomialdistributionforsquarerod.

Parameters:n=6,p=3/4,q=1/4

Mean:μ=np=6·3/4=18/4=4.5

Variance:σ2=npq=6·3/41/4=18/16=1.125

HYPERGEOMETRICDISTRIBUTION

Overview

N=Totalpopulation

S=Populationdefined"success"

F=Populationdefined"failure"

ProabilityDensityFunction(p.d.f.)

Thisisshownmathematicallyas:

Note IftotalnumberofsuccessesS
cumulativedistributionfrunction(CDF)isshownas:

1. Hypergeometricdistributionappliestodiscretesamplestaken

fromafinitepopulationNwithoutreplacement.

2. Anintegerrisoftenusedinplaceofthevariablex.

3. Threeparameters(N,S,andn)specifythisdistribution.

4. Threealternateparameters(N,p,andn),where(p=S/N),may

beusedtodefinethedistribution.

AlternateParametersandProperties

1. Threealternateparameters(N,p,andn),wherep=S/Nis

proportionofsuccess,q=F/N=1-pisproportionoffailures

2. Hypergeometricdistribution(withoutreplacement)

3. Alternativerepresentationofmeanandvariance

Mean:μ=np

1. AspopulationsizeNincreases(N-n)/(N-1)goesto1,and

hypergeometricdistributionapproachesthatofbinomial.

2. Bestunbiasedestimatorofparameterpfromactualdata:p≈

=S/N;populationNnotsamplen.

COMPARISONOFHYPERGEOMETRICANDBINOMIAL

DISTRIBUTIONS

SampleSize

HypergeometricdistributionisbasedonfinitesizepopulationNwith

samplesizentakenwithoutreplacement.Consequence:Probabilities

canvarywithsamplesizen.BinomialandPoissondistributionsassume

eitherafinitepopulationNwithsamplentakenwithreplacementora

verylargepopulationN.Consequence:probabilitiesarethoseof

populationanddonotvarywithsamplesizen.

DiscreteTwoOptions

Bothhypergeometricandbinomialdistributionsarebasedononlytwo

kindsofoutcomes:passorfail.

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CENTRAL LIMIT THEOREM (CLT) - MEAN OF MEANS IS NORMAL (FIGURE 16.28)

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