(1.30)-PowerPoint Ch 11商务统计课件

InferencesAbouttheDifferenceBetweenTwoPopulationProportionsChapter11

ComparisonsInvolvingProportions

andaTestofIndependenceHypothesisTestforProportionsofaMultinomialPopulationTestofIndependenceInferencesAbouttheDifferenceBetween

TwoPopulationProportionsIntervalEstimationofp1-p2HypothesisTestsAboutp1-p2ExpectedValueSamplingDistributionofwhere:n1=sizeofsampletakenfrompopulation1

n2=sizeofsampletakenfrompopulation2StandardDeviation(StandardError)Ifthesamplesizesarelarge,thesamplingdistributionofcanbeapproximatedbyanormalprobabilitydistribution.Thesamplesizesaresufficientlylargeifalloftheseconditionsaremet:n1p1

>5n1(1-p1)>5n2p2

>5n2(1-p2)>5SamplingDistributionofSamplingDistributionofp1–p2IntervalEstimationofp1-p2

IntervalEstimate MarketResearchAssociatesisconductingresearchtoevaluatetheeffectivenessofaclient’snewadvertisingcampaign.Beforethenewcampaignbegan,atelephonesurveyof150householdsinthetestmarketareashowed60households“aware”oftheclient’sproduct.IntervalEstimationofp1-p2

Example:MarketResearchAssociatesThenewcampaignhasbeeninitiatedwithTVandnewspaperadvertisementsrunningforthreeweeks. Asurveyconductedimmediatelyafterthenewcampaignshowed120of250households“aware”oftheclient’sproduct.IntervalEstimationofp1-p2

Example:MarketResearchAssociatesDoesthedatasupportthepositionthattheadvertisingcampaignhasprovidedanincreasedawarenessoftheclient’sproduct?PointEstimatoroftheDifferenceBetween

TwoPopulationProportions=sampleproportionofhouseholds“aware”oftheproductafterthenewcampaign=sampleproportionofhouseholds“aware”oftheproductbeforethenewcampaignp1=proportionofthepopulationofhouseholds“aware”oftheproductafterthenewcampaign

p2=proportionofthepopulationofhouseholds“aware”oftheproductbeforethenewcampaign.08+1.96(.0510).08+.10IntervalEstimationofp1-p2Hence,the95%confidenceintervalforthedifferenceinbeforeandafterawarenessoftheproductis-.02to+.18.For

=.05,z.025=1.96:HypothesisTestsaboutp1-p2HypothesesH0:p1-p2

<0Ha:p1-p2>0Left-tailedRight-tailedTwo-tailedWefocusontestsinvolvingnodifferencebetweenthetwopopulationproportions(i.e.p1=p2)HypothesisTestsaboutp1-p2StandardErrorofwhenp1=p2=p

PooledEstimatorofpwhenp1=p2=p

HypothesisTestsaboutp1-p2TestStatistic

Canweconclude,usinga.05levelofsignificance,thattheproportionofhouseholdsawareoftheclient’sproductincreasedafterthenewadvertisingcampaign?HypothesisTestsaboutp1-p2Example:MarketResearchAssociatesHypothesisTestsaboutp1-p21.Developthehypotheses.

p-ValueandCriticalValueApproachesH0:p1-p2

<0Ha:p1-p2>0p1=proportionofthepopulationofhouseholds“aware”oftheproductafterthenewcampaign

p2=proportionofthepopulationofhouseholds“aware”oftheproductbeforethenewcampaignHypothesisTestsaboutp1-p22.Specifythelevelofsignificance.a=.053.Computethevalueoftheteststatistic.

p-ValueandCriticalValueApproachesHypothesisTestsaboutp1-p25.DeterminewhethertorejectH0.Wecannotconcludethattheproportionofhouseholdsawareoftheclient’sproductincreasedafterthenewcampaign.4.Computethep–value.Forz=1.56,thep–value=.0594Becausep–value>a=.05,wecannotrejectH0.

p–ValueApproachHypothesisTestsaboutp1-p2CriticalValueApproach5.DeterminewhethertorejectH0.Because1.56<1.645,wecannotrejectH0.Fora=.05,z.05=1.6454.Determinethecriticalvalueandrejectionrule.RejectH0ifz

>1.645Wecannotconcludethattheproportionofhouseholdsawareoftheclient’sproductincreasedafterthenewcampaign.HypothesisTestfor

ProportionsofaMultinomialPopulation1.Setupthenullandalternativehypotheses.2.Selectarandomsampleandrecordtheobservedfrequency,fi,foreachofthekcategories.3.AssumingH0istrue,computetheexpectedfrequency,ei,ineachcategorybymultiplyingthecategoryprobabilitybythesamplesize.4.Computethevalueoftheteststatistic.Note:Theteststatistichasachi-squaredistributionwithk–1dfprovidedthattheexpectedfrequenciesare5ormoreforallcategories.fi=observedfrequencyforcategoryiei=expectedfrequencyforcategoryik=numberofcategorieswhere:HypothesisTestfor

ProportionsofaMultinomialPopulationwhere

isthesignificancelevelandtherearek-1degreesoffreedomp-valueapproach:Criticalvalueapproach:RejectH0ifp-value<

a5.Rejectionrule:RejectH0ifHypothesisTestfor

ProportionsofaMultinomialPopulationMultinomialDistributionGoodnessofFitTestExample:FingerLakesHomes(A)FingerLakesHomesmanufacturesfourmodelsofprefabricatedhomes,atwo-storycolonial,alogcabin,asplit-level,andanA-frame.Tohelpinproductionplanning,managementwouldliketodetermineifpreviouscustomerpurchasesindicatethatthereisapreferenceinthestyleselected.ModelColonialLogSplit-LevelA-Frame#Sold

30203515Thenumberofhomessoldofeachmodelfor100salesoverthepasttwoyearsisshownbelow.MultinomialDistributionGoodnessofFitTestExample:FingerLakesHomes(A)HypothesesMultinomialDistributionGoodnessofFitTestwhere:pC=populationproportionthatpurchaseacolonialpL=populationproportionthatpurchasealogcabinpS=populationproportionthatpurchaseasplit-levelpA=populationproportionthatpurchaseanA-frameH0:pC=pL=pS=pA=.25Ha:Thepopulationproportionsarenot

pC=.25,pL=.25,pS=.25,andpA=.25RejectionRule

27.815DoNotRejectH0RejectH0MultinomialDistributionGoodnessofFitTestWith

=.05andk-1=4-1=3degreesoffreedom

RejectH0ifp-value<.05orc2>7.815.ExpectedFrequencies

TestStatisticMultinomialDistributionGoodnessofFitTeste1=.25(100)=25e2=.25(100)=25e3=.25(100)=25e4=.25(100)=25=1+1+4+4=10MultinomialDistributionGoodnessofFitTestConclusionUsingthep-ValueApproachThep-value<

a.Wecanrejectthenullhypothesis.Becausec2

=10isbetween9.348and11.345,theareaintheuppertailofthedistributionisbetween.025and.01.AreainUpperTail.10.05.025.01.005c2Value(df=3)6.2517.8159.34811.34512.838Note:Aprecisep-valuecanbefoundusingMinitaborExcel.ConclusionUsingtheCriticalValueApproachMultinomialDistributionGoodnessofFitTestWereject,atthe.05levelofsignificance,theassumptionthatthereisnohomestylepreference.c2=10>7.815TestofIndependence:ContingencyTables1.Setupthenullandalternativehypotheses.2.Selectarandomsampleandrecordtheobservedfrequency,fij,foreachcellofthecontingencytable.3.Computetheexpectedfrequency,eij,foreachcell.TestofIndependence:ContingencyTables5.Determinetherejectionrule.RejectH0ifp-value<

aor.4.Computetheteststatistic.where

isthesignificanceleveland,withnrowsandmcolumns,thereare(n-1)(m-1)degreesoffreedom.EachhomesoldbyFingerLakesHomescanbeclassifiedaccordingtopriceandtostyle.FingerLakes’managerwouldliketodetermineifthepriceofthehomeandthestyleofthehomeareindependentvariables.ContingencyTable(Independence)TestExample:FingerLakesHomes(B)PriceColonialLogSplit-LevelA-FrameThenumberofhomessoldforeachmodelandpriceforthepasttwoyearsisshownbelow.Forconvenience,thepriceofthehomeislistedaseither$99,000orlessormorethan$99,000.>$99,0001214 163<$99,00018 61912ContingencyTable(Independence)TestExample:FingerLakesHomes(B)HypothesesContingencyTable(Independence)TestH0:PriceofthehomeisindependentofthestyleofthehomethatispurchasedHa:PriceofthehomeisnotindependentofthestyleofthehomethatispurchasedExpectedFrequenciesContingencyTable(Independence)TestPrice

ColonialLogSplit-LevelA-FrameTotal<$99K>$99KTotal3020351510012 1416