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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.

Thevarianceofthemeansisameasureofthespread

ofclustersmeansaboutthetruemean.

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.

7. Randomvariableisadiscreteinteger0≤x≤n.

8. Orderofsuccessnotimportant;combination.

Mean:



Variance:



Example1

Sixtossesofacoin.

Experimentissixtossesofacoin,n=6.

Probabilityofaheadinonetoss,p=1/2.

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

Find:probabilityofgettingexactly2headsin6tosses.



Parameters:n=6cointosses,aheadwithp=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

(2)probabilitydensityissymmmetricaboutthemean.



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

Observationsforp=1/4:(1)themeanisskewedtotheleftofthemidrange,and(2)theprobabilitydensityisnonsymmetricaboutthemean.



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

Observationsforp=3/4:(1)themeanisskewedtotherightofthemidrange,(2)theprobabilitydensityisnonsymmetricaboutthemean.



HYPERGEOMETRICDISTRIBUTION

Overview

TraditionalNotation

N=Totalpopulation

S=Populationdefined"success"

F=Populationdefined"failure"



ProabilityDensityFunction(p.d.f.)

Thisisshownmathematicallyas:



Note IftotalnumberofsuccessesS
cumulativedistributionfrunction(CDF)isshownas:



GeneralComments

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



Comments

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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