商务与经济统计(原书第14版) 课件 第7章　抽样和抽样分布

Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann©2020CengageLearning©2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.1Chapter7-SamplingandSamplingDistributions

2Introduction(1of2)Anelementistheentityonwhichdataarecollected.Apopulationisacollectionofalltheelementsofinterest.Asampleisasubsetofthepopulation.Thesampledpopulationisthepopulationfromwhichthesampleisdrawn.Aframeisalistoftheelementsthatthesamplewillbeselectedfrom.3Introduction(2of2)Thereasonweselectasampleistocollectdatatoansweraresearchquestionaboutapopulation.Thesampleresultsprovideonlyestimatesofthevaluesofthepopulationcharacteristics.Thereasonissimplythatthesamplecontainsonlyaportionofthepopulation.Withpropersamplingmethods,thesampleresultscanprovide“good”estimatesofthepopulationcharacteristics.4SamplingfromaFinitePopulation(1of2)Finitepopulationsareoftendefinedbylistssuchas:OrganizationmembershiprosterCreditcardaccountnumbersInventoryproductnumbersAsimplerandomsampleofsizenfromafinitepopulationofsizeNisasampleselectedsuchthateachpossiblesampleofsizenhasthesameprobabilityofbeingselected.Replacingeachsampledelementbeforeselectingsubsequentelementsiscalledsamplingwithreplacement.Anelementcanappearinthesamplemorethanonce.Samplingwithoutreplacementistheprocedureusedmostoften.Inlargesamplingprojects,computer-generatedrandomnumbersareoftenusedtoautomatethesampleselectionprocess.5SamplingfromaFinitePopulation(2of2)St.Andrew’sCollegereceived900applicationsforadmissionintheupcomingyearfromprospectivestudents.Theapplicantswerenumbered,from1to900,astheirapplicationsarrived.TheDirectorofAdmissionswouldliketoselectasimplerandomsampleof30applicants.Step1: Assignarandomnumbertoeachofthe900applicants. TherandomnumbersgeneratedbyExcel’sRANDfunctionfollowauniformprobabilitydistributionbetween0and1.Step2: Selectthe30applicantscorrespondingtothe30smallestrandomnumbers.6SamplingfromanInfinitePopulation(1of3)Sometimeswewanttoselectasample,butfindthatitisnotpossibletoobtainalistofallelementsinthepopulation.Asaresult,wecannotconstructaframeforthepopulation.Hencewecannotusetherandomnumberselectionprocedure.Mostoftenthissituationoccursinthecaseofinfinitepopulation.7SamplingfromanInfinitePopulation(2of3)Populationsareoftengeneratedbyanongoingprocesswherethereisnoupperlimitonthenumberofunitsthatcanbegenerated.Someexamplesofon-goingprocesseswithinfinitepopulationsare:partsbeingmanufacturedonaproductionlinetransactionsoccurringatabanktelephonecallsarrivingatatechnicalhelpdeskcustomersenteringastore8SamplingfromanInfinitePopulation(3of3)Inthecaseofaninfinitepopulation,wemustselectarandomsampleinordertomakevalidstatisticalinferencesaboutthepopulationfromwhichthesampleistaken.Arandomsamplefromaninfinitepopulationisasampleselectedsuchthatthefollowingconditionsaresatisfied.Eachelementselectedcomesfromthepopulationofinterest.Eachelementisselectedindependently.9PointEstimation(1of4)Pointestimationisaformofstatisticalinference.Inpointestimationweusethedatafromthesampletocomputeavalueofasamplestatisticthatservesasanestimateofapopulationparameter.10PointEstimation(2of4)St.Andrew’sCollegereceived900applicationsfromprospectivestudents.Theapplicationformcontainsavarietyofinformationincludingtheindividual’sScholasticAptitudeTest(SAT)scoreandwhetherornottheindividualdesireson-campushousing.Atameetinginafewhours,theDirectorofAdmissionswouldliketoannouncetheaverageSATscoreandtheproportionofapplicantsthatwanttoliveoncampus,forthepopulationof900applicants.Thedataontheapplicantshavenotyetbeenenteredinthecollege’sdatabase.SotheDirectordecidestoestimatethevaluesofthepopulationparametersofinterestbasedonsamplestatistics.Asampleof30applicantsisselectedusingcomputer-generatedrandomnumbers.11PointEstimation(3of4)

Note:Differentrandomnumberswouldhaveidentifiedadifferentsamplewhichwouldhaveresultedindifferentpointestimates.12PointEstimation(4of4)Onceallthedataforthe900applicantswereenteredinthedatabaseofthecollege,thevaluesofthepopulationparametersofinterestwerecalculated.PopulationMeanSATScore:PopulationStandardDeviationforSATScore:PopulationproportionwantingOn-CampusHousing:13SummaryofPointEstimatesObtainedfromaSimpleRandomSamplePopulationParameterParameterValuePointEstimatorPointEstimateμ=PopulationmeanSATscore16971684σ=Populationstd.deviationforSATscore87.4s=Samplestd.deviationforSATscore85.2p=Populationproportionwantingcampushousing0.720.6714PracticalAdviceThetargetpopulationisthepopulationwewanttomakeinferencesabout.Thesampledpopulationisthepopulationfromwhichthesampleisactuallytaken.Wheneverasampleisusedtomakeinferencesaboutapopulation,weshouldmakesurethatthetargetedpopulationandthesampledpopulationareincloseagreement.15

ProcessofStatisticalInference16

Whentheexpectedvalueofthepointestimatorequalsthepopulationparameter,wesaythepointestimatorisunbiased.17

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Afinitepopulationistreatedasbeinginfiniteifisthefinitepopulationcorrectionfactor.isreferredtoasthestandarderrorofthemean.19

20CentralLimitTheorem

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Example:St.Andrew’sCollege22

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Example:St.Andrew’sCollegeStep1:Calculatethez-valueattheupperendpointoftheinterval.Step2:Findtheareaunderthecurvetotheleftoftheupperendpoint.24

Example:St.Andrew’sCollegeCumulativeProbabilitiesfortheStandardNormalDistributionz.00.01.02.03.04.......5.6915.6950.6985.7019.7054.6.7257.7291.7324.737.7389.7.7580.7611.7642.7673.7704.8.7881.7910.7939.7967.7995.9.8159.8186.8212.8238.826425

Example:St.Andrew’sCollegeStep3:Calculatethez-valueatthelowerendpointoftheinterval.Step4:Findtheareaunderthecurvetotheleftofthelowerendpoint.26

Example:St.Andrew’sCollegeStep5:Calculatetheareaunderthecurvebetweenthelowerandupperendpointsoftheinterval.TheprobabilitythattheestimateofpopulationmeanSATscorewillbebetween1687and1707is:27

Example:St.Andrew’sCollegeSupposeweselectasimplerandomsampleof100applicantsinsteadofthe30originallyconsidered.

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Example:St.Andrew’sCollege29

Example:St.Andrew’sCollege30

Example:St.Andrew’sCollege31

MakingInferencesaboutaPopulationProportion32

where:p=thepopulationproportion33

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Example:St.Andrew’sCollegeRecallthat72%oftheprospectivestudentsapplyingtoSt.Andrew’sCollegedesireon-campushousing.Whatistheprobabilitythatasimplerandomsampleof30applicantswillprovideanestimateofthepopulationproportionofapplicantdesiringon-campushousingthatiswithinplusorminus.05oftheactualpopulationproportion?36

Forourexample,withn=30andp=.72,thenormaldistributionisanacceptableapproximationbecause37

Example:St.Andrew’sCollegeStep1:Calculatethez-valueattheupperendpointoftheinterval.Step2:Findtheareaunderthecurvetotheleftoftheupperendpoint.38

Example:St.Andrew’sCollegeCumulativeProbabilitiesfortheStandardNormalDistributionz.00.01.02.03.04.......5.6915.6950.6985.7019.7054.6.7257.7291.7324.7387.7389.7.7580.7611.7642.7673.7704.8.7881.7910.7939.7967.7995.9.8159.8186.8212.8238.8264......39

Example:St.Andrew’sCollegeStep3:Calculatethez-valueatthelowerendpointoftheinterval.Step4:Findtheareaunderthecurvetotheleftofthelowerendpoint.40

41OtherSamplingMethodsStratifiedRandomSamplingClusterSamplingSystematicSamplingConvenienceSamplingJudgmentSampling42StratifiedRandomSamplingThepopulationisfirstdividedintogroupsofelementscalledstrata.Eachelementinthepopulationbelongstooneandonlyonestratum.Bestresultsareobtainedwhentheelementswithineachstratumareasmuchalikeaspossible(i.e.,ahomogeneousgroup).43StratifiedRandomSampling,Part2Asimplerandomsampleistakenfromeachstratum.Formulasareavailableforcombiningthestratumsampleresultsintoonepopulationparameterestimate.Advantage:Ifstrataarehomogeneous,thismethodprovidesresultsthatareas“precise”assimplerandomsamplingbutwithasmallertotalsamplesize.Example:Thebasisforformingthestratamightbedepartment,location,age,industrytype,andsoon.44ClusterSampling(1of2)Thepopulationisfirstdividedintoseparategroupsofelementscalledclusters.Ideally,eachclusterisarepresentativesmall-scaleversionofthepopulation(i.e.,heterogeneousgroup).Asimplerandomsampleoftheclustersisthentaken.Allelementswithineachsampled(chosen)clusterformthesample.45ClusterSampling(2of2)Example:Aprimaryapplicationisareasampling,whereclustersarecityblocksorotherwell-definedareas.Advantage:Thecloseproximityofelementscanbecosteffective(i.e.,manysampleobservationscanbeobtainedinashorttime).Disadvantage:Thismethodgenerallyrequiresalargertotalsamplesizethansimpleorstratifiedrandomsampling.46SystematicSampling(1of2)IfasamplesizeofnisdesiredfromapopulationcontainingNelements,wemightsampleoneelementforeveryN/nelementsinthepopulation.WerandomlyselectoneofthefirstN/nelementsfromthepopulationlist.WethenselecteveryN/nthelementthatfollowsinthepopulationlist.47SystematicSampling(2of2)Thismethodhasthepropertiesofasimplerandomsample,especiallyifthelistofthepopulationelementsisarandomordering.Advantage:Thesampleusuallywillbeeasiertoidentifythanitwouldbeifsimplerandomsamplingwereused.Example:Selectingevery100thlistinginatelephonebookafterthefirstrandomlyselectedlisting.48ConvenienceSamplingItisanonprobabilitysamplingtechnique.Itemsareincludedinthesamplewithoutknownprobabilitiesofbeingselected.Thesample