商务与经济统计第7章课件

StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter7

SamplingandSamplingDistributionsSamplingDistributionofIntroductiontoSamplingDistributionsPointEstimationSelectingaSampleOtherSamplingMethodsSamplingDistributionof

PropertiesofPointEstimatorsChapter7

SamplingandSamplinIntroductionApopulationisacollectionofalltheelementsof

interest.Asampleisasubsetofthepopulation.Anelementistheentityonwhichdataarecollected.

Aframeisalistoftheelementsthatthesamplewillbeselectedfrom.

Thesampledpopulationisthepopulationfromwhichthesampleisdrawn.IntroductionApopulationisaThesampleresultsprovideonlyestimatesofthevaluesofthepopulationcharacteristics.Withpropersamplingmethods,thesampleresultscanprovide“good”estimatesofthepopulationcharacteristics.IntroductionThereasonissimplythatthesamplecontainsonlyaportionofthepopulation.

Thereasonweselectasampleistocollectdatatoansweraresearchquestionaboutapopulation.ThesampleresultsprovideonSelectingaSampleSamplingfromaFinitePopulationSamplingfromanInfinitePopulationSelectingaSampleSamplingfroSamplingfromaFinitePopulationFinitepopulationsareoftendefinedbylistssuchas:OrganizationmembershiprosterCreditcardaccountnumbersInventoryproductnumbersAsimplerandomsampleofsizenfromafinite

populationofsizeNisasampleselectedsuchthat

eachpossiblesampleofsizenhasthesameprobability

ofbeingselected.SamplingfromaFinitePopulatInlargesamplingprojects,computer-generated

randomnumbersareoftenusedtoautomatethesampleselectionprocess.

Samplingwithoutreplacementistheprocedureusedmostoften.Replacingeachsampledelementbeforeselectingsubsequentelementsiscalledsamplingwith

replacement.SamplingfromaFinitePopulationInlargesamplingprojects, St.Andrew’sCollegereceived900applicationsforadmissionintheupcomingyearfromprospectivestudents.Theapplicantswerenumbered,from1to900,astheirapplicationsarrived.TheDirectorofAdmissionswouldliketoselectasimplerandomsampleof30applicants.Example:St.Andrew’sCollegeSamplingfromaFinitePopulation St.Andrew’sCollegereceiveTherandomnumbersgeneratedbyExcel’s

RANDfunctionfollowauniformprobabilitydistributionbetween0and1.Step1:Assignarandomnumbertoeachofthe900applicants.Step2:Selectthe30applicantscorrespondingtothe30smallestrandomnumbers.SamplingfromaFinitePopulationExample:St.Andrew’sCollegeTherandomnumbersgeneratedSamplingfromanInfinitePopulation

Asaresult,wecannotconstructaframeforthe

population.

Sometimeswewanttoselectasample,butfinditisnotpossibletoobtainalistofallelementsinthe

population.

Hence,wecannotusetherandomnumberselectionprocedure.

Mostoftenthissituationoccursininfinitepopulationcases.SamplingfromanInfinitePopuPopulationsareoftengeneratedbyanongoingprocess

wherethereisnoupperlimitonthenumberofunitsthatcanbegenerated.SamplingfromanInfinitePopulationSomeexamplesofon-goingprocesses,withinfinitepopulations,are:partsbeingmanufacturedonaproductionlinetransactionsoccurringatabanktelephonecallsarrivingatatechnicalhelpdeskcustomersenteringastorePopulationsareoftengenerateSamplingfromanInfinitePopulationArandomsamplefromaninfinitepopulationisasampleselectedsuchthatthefollowingconditionsaresatisfied.Eachelementselectedcomesfromthepopulationofinterest.Eachelementisselectedindependently.

Inthecaseofaninfinitepopulation,wemustselectarandomsampleinordertomakevalidstatisticalinferencesaboutthepopulationfromwhichthesampleistaken.SamplingfromanInfinitePopu

sisthepointestimatorofthepopulationstandarddeviation.Inpointestimationweusethedatafromthesampletocomputeavalueofasamplestatisticthatservesasanestimateofapopulationparameter.PointEstimationWereferto

asthepointestimatorofthepopulationmean.isthepointestimatorofthepopulationproportionp.

Pointestimationisaformofstatisticalinference.sisthepointestimatoroft RecallthatSt.Andrew’sCollegereceived900applicationsfromprospectivestudents.Theapplicationformcontainsavarietyofinformationincludingtheindividual’sScholasticAptitudeTest(SAT)scoreandwhetherornottheindividualdesireson-campushousing.Example:St.Andrew’sCollegePointEstimationAtameetinginafewhours,theDirectorofAdmissionswouldliketoannouncetheaverageSATscoreandtheproportionofapplicantsthatwanttoliveoncampus,forthepopulationof900applicants. RecallthatSt.Andrew’sColPointEstimationExample:St.Andrew’sCollegeHowever,thenecessarydataontheapplicantshavenotyetbeenenteredinthecollege’scomputerizeddatabase.So,theDirectordecidestoestimatethevaluesofthepopulationparametersofinterestbasedonsamplestatistics.Thesampleof30applicantsisselectedusingcomputer-generatedrandomnumbers.PointEstimationExample:St.asPointEstimatorofasPointEstimatorofpPointEstimationNote:

Differentrandomnumberswouldhaveidentifiedadifferentsamplewhichwouldhaveresultedindifferentpointestimates.sasPointEstimatorofasPointEstimatorofPopulationMeanSATScorePopulationStandardDeviationforSATScorePopulationProportionWantingOn-CampusHousingOnceallthedataforthe900applicantswereenteredinthecollege’sdatabase,thevaluesofthepopulationparametersofinterestwerecalculated.PointEstimationPopulationMeanSATScorePopulPopulationParameterPointEstimatorPointEstimateParameterValuem=PopulationmeanSATscore10901097s=Populationstd.deviationforSATscore80

s=Samplestd.deviationforSATscore75.2p=Populationpro-portionwantingcampushousing.72.68SummaryofPointEstimatesObtainedfromaSimpleRandomSample

=SamplemeanSATscore

=Samplepro-portionwantingcampushousingPopulationPointPointParametermPracticalAdviceThetargetpopulationisthepopulationwewanttomakeinferencesabout.Wheneverasampleisusedtomakeinferencesaboutapopulation,weshouldmakesurethatthetargetedpopulationandthesampledpopulationareincloseagreement.Thesampledpopulationisthepopulationfromwhichthesampleisactuallytaken.PracticalAdviceThetargetpoProcessofStatisticalInference

Thevalueofisusedtomakeinferencesaboutthevalueofm.Thesampledataprovideavalueforthesamplemean.Asimplerandomsampleofnelementsisselectedfromthepopulation.Populationwithmeanm=?SamplingDistributionofProcessofStatisticalInferenThesamplingdistributionofistheprobabilitydistributionofallpossiblevaluesofthesamplemean.SamplingDistributionofwhere:

=thepopulationmeanE()=ExpectedValueofWhentheexpectedvalueofthepointestimatorequalsthepopulationparameter,wesaythepointestimatorisunbiased.ThesamplingdistributionSamplingDistributionofWewillusethefollowingnotationtodefinethestandarddeviationofthesamplingdistributionof.s=thestandarddeviationofs=thestandarddeviationofthepopulationn=thesamplesizeN=thepopulationsizeStandardDeviationofSamplingDistributionofSamplingDistributionofFinitePopulationInfinitePopulationisreferredtoasthestandarderrorofthemean.Afinitepopulationistreatedasbeinginfiniteifn/N

<.05.isthefinitepopulation

correctionfactor.StandardDeviationofSamplingDistributionofFiWhenthepopulationhasanormaldistribution,thesamplingdistributionofisnormallydistributedforanysamplesize.Incaseswherethepopulationishighlyskewedoroutliersarepresent,samplesofsize50maybeneeded.Inmostapplications,thesamplingdistributionofcanbeapproximatedbyanormaldistributionwheneverthesampleissize30ormore.SamplingDistributionofWhenthepopulationhasanormSamplingDistributionofThesamplingdistributionofcanbeusedtoprovideprobabilityinformationabouthowclosethesamplemeanistothepopulationmeanm

.SamplingDistributionofThCentralLimitTheoremWhenthepopulationfromwhichweareselectingarandomsampledoesnothaveanormaldistribution,thecentrallimittheoremishelpfulinidentifyingtheshapeofthesamplingdistributionof.

CENTRALLIMITTHEOREMInselectingrandomsamplesofsizenfromapopulation,thesamplingdistributionofthesamplemeancanbeapproximatedbyanormaldistributionasthesamplesizebecomeslarge.CentralLimitTheoremWhenSamplingDistributionofforSATScoresExample:St.Andrew’sCollegeSamplingDistributionofSamplingExample:St.Andrew’sWhatistheprobabilitythatasimplerandomsampleof30applicantswillprovideanestimateofthepopulationmeanSATscorethatiswithin+/-10oftheactualpopulationmean?Example:St.Andrew’sCollegeSamplingDistributionofInotherwords,whatistheprobabilitythatwillbebetween1080and1100?WhatistheprobabilitytStep1:Calculatethez-valueattheupperendpointof theinterval.z=(1100-1090)/14.6=.68P(z

<.68)=.7517Step2:Findtheareaunderthecurvetotheleftofthe

upperendpoint.SamplingDistributionofExample:St.Andrew’sCollegeStep1:Calculatethez-valueCumulativeProbabilitiesfortheStandardNormalDistributionSamplingDistributionofExample:St.Andrew’sCollegeCumulativeProbabilitiesforSa10901100Area=.7517SamplingDistributionofExample:St.Andrew’sCollegeSamplingDistributionofforSATScores10901100Area=.7517SamplingDStep3:Calculatethez-valueatthelowerendpointof theinterval.Step4:Findtheareaunderthecurvetotheleftofthe

lowerendpoint.z=(1080-1090)/14.6=-.68P(z

<-.68)=.2483SamplingDistributionofExample:St.Andrew’sCollegeStep3:Calculatethez-valueSamplingDistributionof

forSATScores10801090Area=.2483Example:St.Andrew’sCollegeSamplingDistributionofforSATScoresSamplingDistributionoffSamplingDistributionof

forSATScoresStep5:Calculatetheareaunderthecurvebetween thelowerandupperendpointsoftheinterval.P(-.68<

z

<.68)=P(z

<.68)-

P(z

<-.68)=.7517-.2483=.5034TheprobabilitythatthesamplemeanSATscorewillbebetween1080and1100is:P(1080<

<1100)=.5034Example:St.Andrew’sCollegeSamplingDistributionoff110010801090SamplingDistributionof

forSATScoresArea=.5034Example:St.Andrew’sCollegeSamplingDistributionofforSATScores110010801090SamplingDistributRelationshipBetweentheSampleSizeandtheSamplingDistributionofSupposeweselectasimplerandomsampleof100applicantsinsteadofthe30originallyconsidered.

E()=mregardlessofthesamplesize.Inour

example,E()remainsat1090.Wheneverthesamplesizeisincreased,thestandarderrorofthemeanisdecreased.Withtheincreaseinthesamplesizeton=100,thestandarderrorofthemeanisdecreasedfrom14.6to:Example:St.Andrew’sCollegeRelationshipBetweentheSamplRelationshipBetweentheSampleSizeandtheSamplingDistributionofWithn=30,Withn=100,Example:St.Andrew’sCollegeRelationshipBetweentheSamplRecallthatwhenn=30,P(1080<

<1100)=.5034.RelationshipBetweentheSampleSizeandtheSamplingDistributionofWefollowthesamestepstosolveforP(1080<

<1100)whenn=100asweshowedearlierwhen

n=30.Now,withn=100,P(1080<

<1100)=.7888.Becausethesamplingdistributionwithn=100hasasmallerstandarderror,thevaluesofhavelessvariabilityandtendtobeclosertothepopulationmeanthanthevaluesofwithn=30.Example:St.Andrew’sCollegeRecallthatwhenn=30,P(RelationshipBetweentheSampleSizeandtheSamplingDistributionof110010801090Area=.7888Example:St.Andrew’sCollegeSamplingDistributionofforSATScoresRelationshipBetweentheSampl

Asimplerandomsampleofnelementsisselectedfromthepopulation.Populationwithproportionp=?MakingInferencesaboutaPopulationProportionThesampledataprovideavalueforthesampleproportion.Thevalueofisusedtomakeinferencesaboutthevalueofp.SamplingDistributionofAsimplerandomsamplePopulaSamplingDistributionofwhere:

p=thepopulationproportionThesamplingdistributionofistheprobabilitydistributionofallpossiblevaluesofthesampleproportion.ExpectedValueofSamplingDistributionofwhere:isreferredtoasthestandarderrorof

theproportion.SamplingDistributionofFinitePopulationInfinitePopulationStandardDeviationofisthefinitepopulationcorrectionfactor.isreferredtoastheFormoftheSamplingDistributionofThesamplingdistributionofcanbeapproximatedbyanormaldistributionwheneverthesamplesizeislargeenoughtosatisfythetwoconditions:...becausewhentheseconditionsaresatisfied,theprobabilitydistributionofxinthesampleproportion,=x/n,canbeapproximatedbynormaldistribution(andbecausenisaconstant).np

>5n(1–p)>5andFormoftheSamplingDistributRecallthat72%oftheprospectivestudentsapplyingtoSt.Andrew’sCollegedesireon-campushousing.Example:St.Andrew’sCollegeSamplingDistributionofWhatistheprobabilitythatasimplerandomsampleof30applicantswillprovideanestimateofthepopulationproportionofapplicantdesiringon-campushousingthatiswithinplusorminus.05oftheactualpopulationproportion?Recallthat72%oftheproForourexample,withn=30andp=.72,thenormaldistributionisanacceptableapproximationbecause:n(1-p)=30(.28)=8.4>5andnp=30(.72)=21.6>5SamplingDistributionofExample:St.Andrew’sCollegeForourexample,withn=SamplingDistributionofSamplingDistributionofExample:St.Andrew’sCollegeSamplingSamplingDistributionStep1:Calculatethez-valueattheupper

endpoint

oftheinterval.z=(.77-.72)/.082=.61P(z

<.61)=.7291Step2:Findtheareaunderthecurvetotheleftof

theupper

endpoint.SamplingDistributionofExample:St.Andrew’sCollegeStep1:Calculatethez-valueCumulativeProbabilitiesfortheStandardNormalDistributionSamplingDistributionofExample:St.Andrew’sCollegeCumulativeProbabilitiesforSa.77.72Area=.7291SamplingDistributionofSamplingDistributionofExample:St.Andrew’sCollege.77.72Area=.7291SamplingSamStep3:Calculatethez-valueatthelowerendpointof theinterval.Step4:Findtheareaunderthecurvetotheleftofthe

lowerendpoint.z=(.67-.72)/.082=-.61P(z

<-.61)=.2709SamplingDistributionofExample:St.Andrew’sCollegeStep3:Calculatethez-value.67.72Area=.2709SamplingDistributionofSamplingDistributionofExample:St.Andrew’sCollege.67.72Area=.2709SamplingSampP(.67<

<.77)=.4582Step5:Calculatetheareaunderthecurvebetween thelowerandupperendpointsoftheinterval.P(-.61<

z

<.61)=P(z

<.61)-

P(z

<-.61)=.7291-.2709=.4582Theprobabilitythatthesampleproportionofapplicantswantingon-campushousingwillbewithin+/-.05oftheactualpopulationproportion:SamplingDistributionofExample:St.Andrew’sCollegeP(.67<<.77)=.4582Step.77.67.72Area=.4582SamplingDistributionofSamplingDistributionofExample:St.Andrew’sCollege.77.67.72Area=.4582SamplingSPropertiesofPointEstimatorsBeforeusingasamplestatisticasapointestimator,statisticianschecktoseewhetherthesamplestatistichasthefollowingpropertiesassociatedwithgoodpointestimators.UnbiasedEfficiencyConsistencyPropertiesofPointEstimatorsPropertiesofPointEstimatorsUnbiased

Iftheexpectedvalueofthesamplestatisticisequaltothepopulationparameterbeingestimated,thesamplestatisticissaidtobeanunbiasedestimatorofthepopulationparameter.PropertiesofPointEstimatorsPropertiesofPointEstimatorsEfficiency

Giventhechoiceoftwounbiasedestimatorsofthesamepopulationparameter,wewouldprefertousethepointestimatorwiththesmallerstandarddeviation,sinceittendstoprovideestimatesclosertothepopulationparameter. Thepointestimatorwiththesmallerstandarddeviationissaidtohavegreaterrelativeefficiencythantheother.PropertiesofPointEstimatorsPropertiesofPointEstimatorsConsistencyApointestimatorisconsistentifthevaluesofthepointestimatortendtobecomeclosertothepopulationparameterasthesamplesizebecomeslarger.Inotherwords,alargesamplesizetendstoprovideabetterpointestimatethanasmallsamplesize.PropertiesofPointEstimatorsOtherSamplingMethodsStratifiedRandomSamplingClusterSamplingSystematicSamplingConvenienceSamplingJudgmentSamplingOtherSamplingMethodsStratifiThepopulationisfirstdividedintogroupsofelementscalledstrata.StratifiedRandomSamplingEachelementinthepopulationbelongstooneandonlyonestratum.Bestresultsareobtainedwhentheelementswithineachstratumareasmuchalikeaspossible(i.e.ahomogeneousgroup).ThepopulationisfirstdividStratifiedRandomSamplingAsimplerandomsampleistakenfromeachstratum.Formulasareavailableforcombiningthestratumsampleresultsintoonepopulationparameterestimate.

Advantage:Ifstrataarehomogeneous,thismethodisas“precise”assimplerandomsamplingbutwithasmallertotalsamplesize.

Example:Thebasisforformingthestratamightbedepartment,location,age,industrytype,andsoon.StratifiedRandomSamplingAsClusterSamplingThepopulationisfirstdividedintoseparategroupsofelementscalledclusters.Ideally,eachclusterisarepresentativesmall-scaleversionofthepopulation(i.e.heterogeneousgroup).Asimplerandomsampleoftheclustersisthentaken.Allelementswithineachsampled(chosen)clusterformthesample.ClusterSamplingThepopulatioClusterSampling

Advantage:Thecloseproximityofelementscanbecosteffective(i.e.manysampleobservationscanbeobtainedinashorttime).

Disadvantage:Thismethodgenerallyrequiresalargertotalsamplesizethansimpleorstratifiedrandomsampling.

Example:Aprimaryapplicationisareasampling,whereclustersarecityblocksorotherwell-definedareas.ClusterSamplingAdvantage:TSystematicSamplingIfasamplesizeofnisdesiredfromapopulationcontainingNelements,wemightsampleoneelementforeveryn/Nelementsinthepopulation.Werandomlyselectoneofthefirstn/Nelementsfromthepopulationlist.Wethenselecteveryn/Nthelementthatfollowsinthepopulationlist.SystematicSamplingIfasamplSystematicSamplingThismethodhasthepropertiesofasimplerandomsample,especiallyifthelistofthepopulationelementsisarandomordering.

Advantage:Thesampleusuallywillbeeasiertoidentifythanitwouldbeifsimplerandomsamplingwereused.

Example:Selectingevery100thlistinginatelephonebookafterthefirstrandomlyselectedlistingSystematicSamplingThismethoConvenienceSamplingItisanonprobabilitysamplingtechnique.Itemsareincludedinthesamplewithoutknownprobabilitiesofbeingselected.

Example:Aprofessorconductingresearchmightusestudentvolunteerstoconstituteasample.Thesampleisidentifiedprimarilybyconvenience.ConvenienceSamplingItisan

Advantage:Sampleselectionanddatacollectionarerelativelyeasy.

Disadvantage:Itisimpossibletodeterminehowrepresentativeofthepopulationthesampleis.ConvenienceSamplingAdvantage:SampleselectionJudgmentSamplingThepersonmostknowledgeableonthesubjectofthestudyselectselementsofthepopulationthatheorshefeelsaremostrepresentativeofthepopulation.Itisanonprobabilitysamplingtechnique.

Example:Areportermightsamplethreeorfoursenators,judgingthemasreflectingthegeneralopinionofthesenate.JudgmentSamplingThepersonmJudgmentSampling

Advantage:Itisarelativelyeasywayofselectingasample.

Disadvantage:Thequalityofthesampleresultsdependsonthejudgmentofthepersonselectingthesample.JudgmentSamplingAdvantage:Recommendation

Itisrecommendedthatprobabilitysamplingmethods(simplerandom,stratified,cluster,orsystematic)beused.

Forthesemethods,formulasareavailableforevaluatingthe“goodness”ofthesampleresultsintermsoftheclosenessoftheresultstothepopulationparametersbeingestimated.

Anevaluationofthegoodnesscannotbemadewithnon-probability(convenienceorjudgment)samplingmethods.RecommendationItisrecommendEndofChapter7EndofChapter7StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter7

SamplingandSamplingDistributionsSamplingDistributionofIntroductiontoSamplingDistributionsPointEstimationSelectingaSampleOtherSamplingMethodsSamplingDistributionof

PropertiesofPointEstimatorsChapter7

SamplingandSamplinIntroductionApopulationisacollectionofalltheelementsof

interest.Asampleisasubsetofthepopulation.Anelementistheentityonwhichdataarecollected.

Aframeisalistoftheelementsthatthesamplewillbeselectedfrom.

Thesampledpopulationisthepopulationfromwhichthesampleisdrawn.IntroductionApopulationisaThesampleresultsprovideonlyestimatesofthevaluesofthepopulationcharacteristics.Withpropersamplingmethods,thesampleresultscanprovide“good”estimatesofthepopulationcharacteristics.IntroductionThereasonissimplythatthesamplecontainsonlyaportionofthepopulation.

Thereasonweselectasampleistocollectdatatoansweraresearchquestionaboutapopulation.ThesampleresultsprovideonSelectingaSampleSamplingfromaFinitePopulationSamplingfromanInfinitePopulationSelectingaSampleSamplingfroSamplingfromaFinitePopulationFinitepopulationsareoftendefinedbylistssuchas:OrganizationmembershiprosterCreditcardaccountnumbersInventoryproductnumbersAsimplerandomsampleofsizenfromafinite

populationofsizeNisasampleselectedsuchthat

eachpossiblesampleofsizenhasthesameprobability

ofbeingselected.SamplingfromaFinitePopulatInlargesamplingprojects,computer-generated

randomnumbersareoftenusedtoautomatethesampleselectionprocess.

Samplingwithoutreplacementistheprocedureusedmostoften.Replacingeachsampledelementbeforeselectingsubsequentelementsiscalledsamplingwith

replacement.SamplingfromaFinitePopulationInlargesamplingprojects, St.Andrew’sCollegereceived900applicationsforadmissionintheupcomingyearfromprospectivestudents.Theapplicantswerenumbered,from1to900,astheirapplicationsarrived.TheDirectorofAdmissionswouldliketoselectasimplerandomsampleof30applicants.Example:St.Andrew’sCollegeSamplingfromaFinitePopulation St.Andrew’sCollegereceiveTherandomnumbersgeneratedbyExcel’s

RANDfunctionfollowauniformprobabilitydistributionbetween0and1.Step1:Assignarandomnumbertoeachofthe900applicants.Step2:Selectthe30applicantscorrespondingtothe30smallestrandomnumbers.SamplingfromaFinitePopulationExample:St.Andrew’sCollegeTherandomnumbersgeneratedSamplingfromanInfinitePopulation

Asaresult,wecannotconstructaframeforthe

population.

Sometimeswewanttoselectasample,butfinditisnotpossibletoobtainalistofallelementsinthe

population.

Hence,wecannotusetherandomnumberselectionprocedure.

Mostoftenthissituationoccursininfinitepopulationcases.SamplingfromanInfinitePopuPopulationsareoftengeneratedbyanongoingprocess

wherethereisnoupperlimitonthenumberofunitsthatcanbegenerated.SamplingfromanInfinitePopulationSomeexamplesofon-goingprocesses,withinfinitepopulations,are:partsbeingmanufacturedonaproductionlinetransactionsoccurringatabanktelephonecallsarrivingatatechnicalhelpdeskcustomersenteringastorePopulationsareoftengenerateSamplingfromanInfinitePopulationArandomsamplefromaninfinitepopulationisasampleselectedsuchthatthefollowingconditionsaresatisfied.Eachelementselectedcomesfromthepopulationofinterest.Eachelementisselectedindependently.

Inthecaseofaninfinitepopulation,wemustselectarandomsampleinordertomakevalidstatisticalinferencesaboutthepopulat