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1、APStatisticsSamplingDistributionofaSampleMean样本均值的抽样分布Beijing21stCenturyInternationalSchool北京二十一世纪国际学校Grade12高三年级YiningLiu刘亦柠“不明于计数，而欲举大事，犹无舟楫而欲经于水险也。”--《管子》ObjectivesDistinguishPopulationDistribution,SamplingDistributionandStatisticDistributionKnowthesignifi
2、canceofmeanandvarianceofSamplingDistributionofaSampleMean,findtherelationshipfromtheparametersofpopulation.ExploreCentralLimitTheorem(inExperiment)Summarizetheadvantagesofsamplingandstatisticdistribution2Review: MeanandVarianceofDiscreteRandomVariableSupposet
3、hatXisadiscreterandomvariablewhoseprobabilitydistributionisValue:x1x2x3…Probability:p1p2p3…Tofindthemean(expectedvalue)ofX,ThevarianceofXis:ProblemSolvingQualitycontrolofNongfuSpringontheassemblyline.Examinetheaveragelevelofbacteriaineachbottle.Question1:Hows
4、houldweexamineit?Difficulties?Question2:Howshouldwedrawasample?Question3:Doesthesamplerepresentthepopulationverywell?PrologueIn1802,Pierre-Simon Laplace,aninfluentialFrenchscholarfirstusethesampletoestimatethenumberofpeopleinafamilyinFrance.Thisistheearliestr
5、ecordofusingsampleinference.SamplingVariabilityDifferentrandomsamplesyielddifferentstatistics.Thisbasicfactiscalledsamplingvariability:thevalueofastatisticvariesinrepeatedrandomsampling.Tomakesenseofsamplingvariability,weask,“Whatwouldhappenifwetookmanymanyma
6、nysamples?”PopulationSampleSampleSampleSampleSampleSampleSampleSample?Experiment1:Population:{1,1,2,2,3,3,4,4}Step1:CompletetheProbabilitymodelandpopulationdistribution:x=thenumberyougetfromonerandomchosenPopulationDistributionx1234P(x)P(x)1/41/41/41/4Populat
7、ionmeanPopulationvariance2.51.25Step2:Drawthesample(n=2):Ifwerandomlyselect2numbersfromthepopulation(duplicationisallowed),allthepossibleeventsetsare:A={{1,1},{1,2},{},{},{},{},{},{},{},{},{},{},{},{},{},{}}Becauseallindividualrandomphenomenonhasequallylikely
8、outcomethattheprobabilityis1/4,andthenumberselectionsareindependentfromeachtrail,sotheprobabilityoftheoccurrenceofeverypossibleeventis.AllsamplemeanofapossibleSRS:11,31,42,12,22,32,43,13,
