人大陈卫教授手把手教你统计Lec-7-Sampling-Distribution.ppt

1 Lecture7SamplingDistribution 2 Thislecturecovers PopulationParametersSampleStatisticsSamplingDistribution 3 Introduction Researchersinthesocialsciencesalmostneverhaveenoughtimeormoneytocollectinformationabouttheentirepopulationthatintereststhem forexample everyoneinBeijing oreveneveryoneinChina Fortunately wecanlearnalotaboutapopulationifwecarefullyselectasubset 子集 ofit Thissubsetiscalledasample 4 Throughtheprocessofsampling selectingasubsetfromthepopulation weattempttogeneralizetothecharacteristicsofthepopulationbasedonwhatwelearnfromthesample Parameter measuresusedtodescribethedistributionofthepopulation populationmeanpopulationproportionpopulationstandarddeviation 5 Weusethetermsamplestatisticwhenreferringtoacorrespondingcharacteristiccalculatedforthesample samplemeansampleproportionsamplestandarddeviationwerarelyknowthevalueoftheparameter 6 Thegoalofmostresearchistofindthepopulationparameter Researchersusuallyselectasamplefromthepopulationandobtainanestimateofthepopulationparameterfromthesample objectiveofsamplingtheoryandstatisticalinferenceistoprovideestimatesofunknownparametersfromsamplestatisticsthatcanbeeasilycalculated 7 mostimportantconceptsinsamplingtheoryandstatisticalinference samplingdistribution TOhelpsestimatethelikelihoodofsamplestatisticsandenablesustogeneralizefromthesampletothepopulation 8 ThePopulation conceptofthesamplingdistribution let sconsiderasourpopulationthe20individualslistedintheright OurvariableYistheincomeforthe20individualstheparameterwearetryingtoestimateisthemeanincome 9 PopulationParameters populationmeanandstandarddeviation 10 TheSample populationmeanisunknownweestimateitsvaluebydrawingarandomsampleofthreeindividuals N 3 calculatethemeanincomeforthissample 11 SampleStatistic meanforthesample Noticethatoursamplemeandiffersfromthepopulationparameter Thisdiscrepancyisduetosamplingerror 12 SamplingError Samplingerroristhediscrepancybetweensampleestimateofapopulationparameterandtherealpopulationparameter Thesampleerrorforthissampleis 22766 20817 1949 13 AnotherSample anotherrandomsample meanforthissampleis 14 TheDilemma Thesamplingerrorforthissampleis22766 21345 1421 somewhatlessthantheerrorforthefirstsample Dilemma Ifsampleestimatesvaryandifmostestimatesresultinsomesortofsamplingerror howmuchconfidencecanweplaceintheestimate Onwhatbasiscanweinferfromthesampletothepopulation 15 TheSamplingDistribution Theanswertothisdilemmaistouseadeviceknownasthesamplingdistribution Thesamplingdistributionisatheoreticalprobabilitydistribution 抽样分布是理论上的概率分布 ofallpossiblesamplevaluesforthestatisticinwhichweareinterested 16 TheSamplingDistribution Ifweweretodrawallpossiblerandomsamplesofthesamesizefromourpopulationofinterestcomputetheeachstatisticplotthefrequencydistributionforthatstatisticwewouldobtainanapproximationofthesamplingdistribution 17 Everystatistic forexample aproportion amean astandarddeviation hasasamplingdistribution Becauseitincludesallpossiblesamplevalues thesamplingdistributionenablesustocompareoursampleresultwithothersamplevaluesanddeterminethelikelihoodassociatedwiththatresult 18 SamplingDistributionoftheMean Samplingdistributionsaretheoreticaldistributions whichmeansthattheyareneverreallyobserved theyarenonempirical samplingdistributionofthemeanisatheoreticaldistributionofsamplemeansthatwouldobtainedbydrawingfromthepopulationallpossiblesamplesofthesamesize 19 20 Inillustratingtherelationshipbetweenthepopulation thesample andthesamplingdistributions Ihavegeneratedanempiricalsamplingdistributionofthemeanageatfirstmarriage Considerasthepopulationof5006womeninthe1997survey Wedraw500samplesofN 20andcalculatethemeanageatfirstmarriageofeachsample 21 Makeahistogramofthe500means Thisisthesamplingdistributionofthemeanshowingthemeanvaluesandfrequency numberofsamples 22 ThePopulationDistribution 21 4801 2 701 23 500samplesofN 20 21 4481 0 68971 24 samplingdistributioncanbedescribedintermsofitsmeanandstandarddeviation Toobtainedthemeanofthesamplingdistribution addallindividualsamplemeansanddividebythenumberofsamples Thus themeanofthesamplingdistributionofthemeanisactuallythemeanofmeans 25 Themeanofthesamplingdistributionofthemean 抽样分布均值的均值 26 StandardErroroftheMean Thestandarddeviationofthesamplingdistributioniscalledthestandarderrorofthemean itisameasuretodescribehowmuchdispersioninthesamplingdistributionofthemean 均值的抽样分布的离散程度 27 itisequalto thestandarddeviationofthepopulation6ydividedbythesquarerootofthesamplesize n 1 2 28 BasicPropertiesofSamplingDistributionoftheMean 1Regardlessoftheshapeofthepopulationdistribution thesamplingdistributionofthemeanisapproximatelynormal 2onlyafewofthesamplemeanscoincideexactlywiththepopulationmean 3samplingdistributioncentersaroundthepopulationmean 29 variabilityofthesamplingdistributionissmallerthanthevariabilityofthepopulationdistribution Thesepropertiesareevenmorestrikingasthesamplesizeincreases 每次抽样的数量N themeanofthesamplingdistributiongettingclosertothepopulationmeanandstandarderrorofthemeanbecomingsmaller 30 500samplesofN 100 21 4979 0 29517 31 500samplesofN 400 21 4837 0 13838 32 CentralLimitTheorem AsthesamplesizeNincreases thesamplingdistributionofthemeanmoreandmorecloselyresemblesanormaldistribution USU 30ormore 33 34 Thusevenifavariableisnotnormallydistributedinthepopulation themeanofallpossiblesamplemeansofthisvariableisthesameasthepopulationmean 35 ittellsusthatwithsufficientsamplesize thesamplingdistributionofthemeanwillbenormalregardlessoftheshapeofthepopulationdistribution evenwhenthepopulationdistributionisskewedwecanstillassumethatthesamplingdistr