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)

DenominatorDegreesofFreedomAreainUpperTailNumeratorDegreesofFreedomat7NumeratorDegreesofFreedomat8NumeratorDegreesofFreedomat9Numerato