商务与经济统计(原书第14版) 课件 第11、12章　总体方差的统计推断、多个比率的比较、独立性及拟合优度检验

Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann©2020CengageLearning1©2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.Chapter11-InferencesAboutPopulationVariances11.1-InferencesAboutaPopulationVariance11.2-InferencesAboutTwoPopulationsVariances2InferencesAboutaPopulationVarianceAvariancecanprovideimportantdecision-makinginformation.Considertheproductionprocessoffillingcontainerswithaliquiddetergentproduct.Themeanfillingweightisimportant,butalsoisthevarianceofthefillingweights.Byselectingasampleofcontainers,wecancomputeasamplevariancefortheamountofdetergentplacedinacontainer.Ifthesamplevarianceisexcessive,overfillingandunderfillingmaybeoccurringeventhoughthemeaniscorrect.3Chi-SquareDistribution(1of2)Thechi-squaredistributionisbasedonsamplingfromanormalpopulation.Wecanusethechi-squaredistributiontodevelopintervalestimatesandconducthypothesistestsaboutapopulationvariance.4

5Chi-SquareDistribution(2of2)6IntervalEstimationofσ2(1of8)7IntervalEstimationofσ2(2of8)Takingthesquarerootoftheupperandlowerlimitsofthevarianceintervalprovidestheconfidenceintervalforthepopulationstandarddeviation.8IntervalEstimationofσ2(3of8)

Example:Buyer’sDigest(A) Buyer’sDigestratesthermostatsmanufacturedforhometemperaturecontrol.Inarecenttest,10thermostatsmanufacturedbyThermoRitewereselectedandplacedinatestroomthatwasmaintainedatatemperatureof68oF.Wewillusethe10readingsbelowtodevelopa95%confidenceintervalestimateofthepopulationvariance.Thermostat12345678910Temperature67.467.868.269.369.567.068.168.667.967.29IntervalEstimationofσ2(4of8)SelectedValuesfromtheChi-SquareDistributionTableForn–1=10–1=9dfandα=0.05DegreesofFreedom.99AreainUpperTail.975AreainUpperTail.95AreainUpperTail.90AreainUpperTail.10AreainUpperTail.05AreainUpperTail.025AreainUpperTail.01AreainUpperTail50.5540.8311.1451.6109.23611.07012.83215.08660.8721.2371.6352.20410.64512.59214.44916.81271.2391.6902.1672.83312.01714.06716.01318.47581.6472.1802.7333.49013.36215.50717.53520.09092.0882.7003.3254.16814.68416.91919.02321.666102.5583.2473.9404.86515.98718.30720.48323.20910IntervalEstimationofσ2(5of8)Forn–1=10–1=9dfandα=0.0511IntervalEstimationofσ2(6of8)Forn–1=10–1=9dfandα=0.05DegreesofFreedom.99AreainUpperTail.975AreainUpperTail.95AreainUpperTail.90AreainUpperTail.10AreainUpperTail.05AreainUpperTail.025AreainUpperTail.01AreainUpperTail50.5540.8311.1451.6109.23611.07012.83215.08660.8721.2371.6352.20410.64512.59214.44916.81271.2391.6902.1672.83312.01714.06716.01318.47581.6472.1802.7333.49013.36215.50717.53520.09092.0882.7003.3254.16814.68416.91919.02321.666102.5583.2473.9404.86515.98718.30720.48323.20912IntervalEstimationofσ2(7of8)n–1=10–1=9degreesoffreedomandα=0.0513IntervalEstimationofσ2(8of8)Thesamplevariances2providesapointestimateofσ2.A95%confidenceintervalforthepopulationvarianceisgivenby:14HypothesisTestingAboutaPopulationVariance(1of8)15HypothesisTestingAboutaPopulationVariance(2of8)Foreachtypeoftest,

16HypothesisTestingAboutaPopulationVariance(3of8)Example:Buyer’sDigest(B)RecallthatBuyer’sDigestisratingThermoRitethermostats.Buyer’sDigestgivesan“acceptable”ratingtoathermostatwithatemperaturevarianceof0.5orless.Usingthe10readings,wewillconductahypothesistest(witha=0.10)todeterminewhethertheThermoRitethermostat’stemperaturevarianceis“acceptable.”Thermostat12345678910Temperature67.467.868.269.369.567.068.168.667.967.217HypothesisTestingAboutaPopulationVariance(4of8)18HypothesisTestingAboutaPopulationVariance(5of8)Forn–1=10–1=9dfanda=0.10SelectedValuesfromtheChi-SquareDistributionTableDegreesofFreedom.99AreainUpperTail.975AreainUpperTail.95AreainUpperTail.90AreainUpperTail.10AreainUpperTail.05AreainUpperTail.025AreainUpperTail.01AreainUpperTail50.5540.8311.1451.6109.23611.07012.83215.08660.8721.2371.6352.20410.64512.59214.44916.81271.2391.6902.1672.83312.01714.06716.01318.47581.6472.1802.7333.49013.36215.50717.53520.09092.0882.7003.3254.16814.68416.91919.02321.666102.5583.2473.9404.86515.98718.30720.48323.20919HypothesisTestingAboutaPopulationVariance(6of8)RejectionRegion20HypothesisTestingAboutaPopulationVariance(7of8)21HypothesisTestingAboutaPopulationVariance(8of8)Usingthep-Value22InferencesAboutTwoPopulationVariancesWemaywanttocomparethevariancesin:productqualityresultingfromtwodifferentproductionprocesses,temperaturesfortwoheatingdevices,orassemblytimesfortwoassemblymethodsWeusedatacollectedfromtwoindependentrandomsamples,onefrompopulation1andanotherfrompopulation2.Thetwosamplevarianceswillbethebasisformakinginferencesaboutthetwopopulationvariances.23HypothesisTestingAboutaPopulationVariance(1of2)24HypothesisTestingAboutaPopulationVariance(2of2)Foreachtypeoftest,25HypothesisTestingAbouttheVariancesofTwoPopulations(1of5)Example:Buyer’sDigest(C)Buyer’sDigesthasconductedthesametest,asdescribedearlier,onanother10thermostats,thistimemanufacturedbyTempKing.Wewillconductahypothesistestwithα

=0.10toseeifthevariancesareequalforThermoRite’sthermostatsandTempKing’sthermostats.ThermoRiteSampleThermostat12345678910Temperature67.467.868.269.369.567.068.168.667.967.2TempKingSampleThermostat12345678910Temperature67.766.469.270.169.569.768.166.667.367.526HypothesisTestingAbouttheVariancesofTwoPopulations(2of5)HypothesesRejectionRule27HypothesisTestingAbouttheVariancesofTwoPopulations(3of5)

DenominatorDegreesofFreedomAreainUpperTailNumeratorDegreesofFreedomat7NumeratorDegreesofFreedomat8NumeratorDegreesofFreedomat9NumeratorDegreesofFreedomat10NumeratorDegreesofFreedomat158.102.622.592.562.542.468.053.503.443.393.353.228.0254.534.434.364.304.108.016.186.035.915.815.529.102.512.472.442.422.349.053.293.233.183.143.019.0254.204.104.033.963.779.015.615.475.355.264.9628HypothesisTestingAbouttheVariancesofTwoPopulations(4of5)TestStatistic:TempKing’ssamplevarianceis1.768.ThermoRite’ssamplevarianceis0.7.Conclusion:29HypothesisTestingAbouttheVariancesofTwoPopulations(5of5)Determiningandusingthep-ValueAreainUpperTail.10.05.025.01FValueatDegreesoffreedomsubscript1baselineequals9anddegreesoffreedomsubscript2baselineequals92.443.184.035.35

30Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann©2020CengageLearning31©2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.Chapter12-ComparingMultipleProportions,TestofIndependenceandGoodnessofFit12.1-TestingtheEqualityofPopulationProportionsforThreeorMorePopulations12.2-TestofIndependence12.3-GoodnessofFitTest32TestsofGoodnessofFit,Independence,andMultipleProportionsInthischapterweintroducethreeadditionalhypothesis-testingprocedures.Theteststatisticandthedistributionusedarebasedonthechi-square(χ2)distribution.Inallcases,thedataarecategorical.33TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(1of12)Hypotheses:34TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(2of12)

35TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(3of12)Example:FingerLakesHomesFingerLakesHomesmanufacturesthreemodelsofprefabricatedhomes:atwo-storycolonial,alogcabin,andanA-frame.Tohelpinproduct-lineplanning,managementwouldliketocomparethecustomersatisfactionwiththethreehomestyles.p1=proportionlikelytorepurchaseaColonialforthepopulationofColonialownersp2=proportionlikelytorepurchaseaLogCabinforthepopulationofLogCabinownersp3=proportionlikelytorepurchaseanA-FrameforthepopulationofA-Frameowners36TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(4of12)Webeginbytakingasampleofownersfromeachofthethreepopulations.Eachsamplecontainscategoricaldataindicatingwhethertherespondentsarelikelyornotlikelytorepurchasethehome.Herearetheobservedfrequencies(sampleresults)LikelytorepurchaseHomeownerColonialHomeOwnerLogHomeOwnerA-frameTotalYes978380260No381844100Total13510112436037TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(5of12)Next,wedeterminetheexpectedfrequenciesundertheassumptionH0iscorrect.ExpectedfrequenciesundertheassumptionH0istruearecalculatedusingthisformula:Ifasignificantdifferenceexistsbetweentheobservedandexpectedfrequencies,H0canberejected.38TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(6of12)Theexpectedfrequenciesare:LikelytorepurchaseHomeOwnerColonialHomeOwnerLogHomeOwnerA-frameTotalYes97.5072.9489.56260No37.5028.0634.44100Total13510112436039TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(7of12)Next,computethevalueofthechi-squareteststatistic.

40TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(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=8.670041TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(9of12)RejectionRule:42TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(10of12)RejectionRule(usinga=0.05)43TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(11of12)Conclusionusingthep-valueapproach:Areainuppertail.10.05.025.01.005Chisquared(df=2)4.6055.9917.3789.21010.59744TestingtheEqualityofPopulationProportionsforThreeorMorePopulations(12of12)Wehaveconcludedthatthepopulationproportionsforthethreepopulationsofhomeownersarenotequal.Toidentifywherethedifferencesbetweenpopulationproportionsexist,wewillrelyonamultiplecomparisonsprocedure.45MultipleComparisonsProcedure(1of4)Webeginbycomputingthethreesampleproportions.WewilluseamultiplecomparisonprocedureknownastheMarascuiloprocedure.46MultipleComparisonsProcedure(2of4)MarascuiloProcedureWecomputetheabsolutevalueofthepairwisedifferencebetweensampleproportions.47MultipleComparisonsProcedure(3of4)CriticalValuesfortheMarascuiloPairwiseComparisonForeachpairwisecomparison,computeacriticalvalueasfollows:48MultipleComparisonsProcedure(4of4)PairwiseComparisonTestsPairwiseComparisonCVijColonialvs.LogCabin.1033.1329NotsignificantColonialvs.A-Frame.0733.1415NotSignificantLogCabinvs.A-Frame.1766.1405Significant49TestofIndependence(1of7)

50TestofIndependence(2of7)Computetheteststatistic.Determinetherejectionrule.51TestofIndependence(3of7)Example:FingerLakesHomes(B)EachhomesoldbyFingerLakesHomescanbeclassifiedaccordingtopriceandtostyle.FingerLakes’managerwouldliketodetermineifthepriceofthehomeandthestyleofthehomeareindependentvariables.Thenumberofhomessoldforeachmodelandpriceforthepasttwoyearsisshownbelow.Forconvenience,thepriceofthehomeislistedaseitherlessthan$200,000orgreaterthanorequalto$200,000.PriceColonialLogSplit-LevelA-Frame<$200,0001861912≥$200,000121416352TestofIndependence(4of7)HypothesesH0:PriceofthehomeisindependentofthestyleofthehomethatispurchasedHa:PriceofthehomeisnotindependentofthestyleofthehomethatispurchasedExpectedFrequenciesPriceColonialLogSplit-LevelA-frameTotal<$200K186191255≥$200K121416345Total3020351510053TestofIndependence(5of7)RejectionRuleTestStatistic54TestofIndependence(6of7)Conclusionusingthep-valueapproachAreainuppertail.10.05.025.01.005χ2value(df=3)6.2517.8159.34811.34512.83855TestofIndependence(7of7)ConclusionusingthecriticalvalueapproachWerejectatthe0.05levelofsignificancetheassumptionthatthepriceofthehomeisindependentofthestyleofhomethatispurchased.56GoodnessofFitTest:MultinomialProbabilityDistribution(1of3)

57GoodnessofFitTest:MultinomialProbabilityDistribution(2of3)Computethevalueofthe