Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann?2020CengageLearning1?2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.Chapter12-ComparingMultipleProportions,TestofIndependenceandGoodnessofFit12.1-TestingtheEqualityofPopulationProportionsforThreeorMorePopulations12.2-TestofIndependence12.3-GoodnessofFitTest2TestsofGoodnessofFit,Independence,andMultipleProportionsInthischapterweintroducethreeadditionalhypothesis-testingprocedures.Theteststatisticandthedistributionusedarebasedonthechi-square(χ2)distribution.Inallcases,thedataarecategorical.3TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(1of12)Hypotheses:4TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(2of12)

5TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(3of12)Example:FingerLakesHomesFingerLakesHomesmanufacturesthreemodelsofprefabricatedhomes:atwo-storycolonial,alogcabin,andanA-frame.Tohelpinproduct-lineplanning,managementwouldliketocomparethecustomersatisfactionwiththethreehomestyles.p1=proportionlikelytorepurchaseaColonialforthepopulationofColonialownersp2=proportionlikelytorepurchaseaLogCabinforthepopulationofLogCabinownersp3=proportionlikelytorepurchaseanA-FrameforthepopulationofA-Frameowners6TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(4of12)Webeginbytakingasampleofownersfromeachofthethreepopulations.Eachsamplecontainscategoricaldataindicatingwhethertherespondentsarelikelyornotlikelytorepurchasethehome.Herearetheobservedfrequencies(sampleresults)LikelytorepurchaseHomeownerColonialHomeOwnerLogHomeOwnerA-frameTotalYes978380260No381844100Total1351011243607TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(5of12)Next,wedeterminetheexpectedfrequenciesundertheassumptionH0iscorrect.ExpectedfrequenciesundertheassumptionH0istruearecalculatedusingthisformula:Ifasignificantdifferenceexistsbetweentheobservedandexpectedfrequencies,H0canberejected.8TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(6of12)Theexpectedfrequenciesare:LikelytorepurchaseHomeOwnerColonialHomeOwnerLogHomeOwnerA-frameTotalYes97.5072.9489.56260No37.5028.0634.44100Total1351011243609TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(7of12)Next,computethevalueofthechi-squareteststatistic.

10TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(8of12)ComputationoftheChi-SquareTestStatisticLikelytoRepurchaseHomeOwnerObs.Freq.offsubscriptijbaselineExp.Freq.ofesubscriptijbaselineDiff.ofopenparenthesisfsubscriptijbaselineminusesubscriptijbaselineCloseparenthesisSqd.Diff.ofopenparenthesisfsubscriptijbaselineminusesubscriptijbaselineCloseparenthesissquaredSqd.Diff/Exp.Freq.of(fij–eij)2/eijStartfraction,openparenthesisfsubscriptijbaselineminusesubscriptijbaselineCloseparenthesissquared,overesubscriptijbaselineYes

Colonial9797.50-0.500.25000.0026YesLogCab.8372.9410.06101.11421.3862YesA-frame8089.56-9.5691.30861.0196NoColonial3837.500.500.25000.00687NoLogCab.1828.06-10.06101.11423.6041NoA-frame4434.449.5691.30862.6509EmptycellTotal360360Emptycellxsquared=8.670011TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(9of12)RejectionRule:12TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(10of12)RejectionRule(usinga=0.05)13TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(11of12)Conclusionusingthep-valueapproach:Areainuppertail.10.05.025.01.005Chisquared(df=2)4.6055.9917.3789.21010.59714TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(12of12)Wehaveconcludedthatthepopulationproportionsforthethreepopulationsofhomeownersarenotequal.Toidentifywherethedifferencesbetweenpopulationproportionsexist,wewillrelyonamultiplecomparisonsprocedure.15MultipleComparisonsProcedure(1of4)Webeginbycomputingthethreesampleproportions.WewilluseamultiplecomparisonprocedureknownastheMarascuiloprocedure.16MultipleComparisonsProcedure(2of4)MarascuiloProcedureWecomputetheabsolutevalueofthepairwisedifferencebetweensampleproportions.17MultipleComparisonsProcedure(3of4)CriticalValuesfortheMarascuiloPairwiseComparisonForeachpairwisecomparison,computeacriticalvalueasfollows:18MultipleComparisonsProcedure(4of4)PairwiseComparisonTestsPairwiseComparisonCVijColonialvs.LogCabin.1033.1329NotsignificantColonialvs.A-Frame.0733.1415NotSignificantLogCabinvs.A-Frame.1766.1405Significant19TestofIndependence(1of7)

20TestofIndependence(2of7)Computetheteststatistic.Determinetherejectionrule.21TestofIndependence(3of7)Example:FingerLakesHomes(B)EachhomesoldbyFingerLakesHomescanbeclassifiedaccordingtopriceandtostyle.FingerLakes’managerwouldliketodetermineifthepriceofthehomeandthestyleofthehomeareindependentvariables.Thenumberofhomessoldforeachmodelandpriceforthepasttwoyearsisshownbelow.Forconvenience,thepriceofthehomeislistedaseitherlessthan$200,000orgreaterthanorequalto$200,000.PriceColonialLogSplit-LevelA-Frame<$200,0001861912≥$200,000121416322TestofIndependence(4of7)HypothesesH0:PriceofthehomeisindependentofthestyleofthehomethatispurchasedHa:PriceofthehomeisnotindependentofthestyleofthehomethatispurchasedExpectedFrequenciesPriceColonialLogSplit-LevelA-frameTotal<$200K186191255≥$200K121416345Total3020351510023TestofIndependence(5of7)RejectionRuleTestStatistic24TestofIndependence(6of7)Conclusionusingthep-valueapproachAreainuppertail.10.05.025.01.005χ2value(df=3)6.2517.8159.34811.34512.83825TestofIndependence(7of7)ConclusionusingthecriticalvalueapproachWerejectatthe0.05levelofsignificancetheassumptionthatthepriceofthehomeisindependentofthestyleofhomethatispurchased.26GoodnessofFitTest:MultinomialProbabilityDistribution(1of3)

27GoodnessofFitTest:MultinomialProbabilityDistribution(2of3)Computethevalueoftheteststatistic.

2829GoodnessofFitTest:MultinomialProbabilityDistribution(3of3)RejectionRule:MultinomialDistributionGoodnessofFitTest(1of6)Example:FingerLakesHomes(A)FingerLakesHomesmanufacturesfourmodelsofprefabricatedhomes:atwo-storycolonial,alogcabin,asplit-level,andanA-frame.Tohelpinproductionplanning,managementwouldliketodetermineifpreviouscustomerpurchasesindicatethatthereisapreferenceinthestyleselected.Thenumberofhomessoldofeachmodelfor100salesoverthepasttwoyearsisshownbelow.ModelColonialLogSplit-LevelA-Frame#Sold3020351530MultinomialDistributionGoodnessofFitTest(2of6)HypothesesH0:pC=pL=pS=pA=.25Ha:Thepopulati