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1、SamplingDistributionsChapter9AUCh9-2Outline9.1SamplingDistributionoftheMean9.2SamplingDistributionoftheProportion9.3SamplingDistributionoftheDifferencebetweenTwoMeans(9.4FromHeretoInference)AUCh9-3IntroductionInreallifecalculatingparametersofpopulationsisprohibitive
2、becausepopulationsareverylarge.Ratherthaninvestigatingthewholepopulation,wetakeasample,calculateastatisticrelatedtotheparameterofinterest,andmakeaninference.Thesamplingdistributionofthestatisticisthetoolthattellsushowcloseisthestatistictotheparameter.AUCh9-49.1Sampl
3、ingDistributionofthe(sample)Mean例題丟一個骰子無限次隨機變數X表示骰子出現的點數X的機率分配(表與圖),平均數,變異數為：E(X)=m=1(1/6)+2(1/6)+3(1/6)+…+6(1/6)=3.5s2=V(X)=(1-3.5)2(1/6)+(2-3.5)2(1/6)+…+(6-3.5)2(1/6)=2.92x123456P(x)1/61/61/61/61/61/61/6123456AUCh9-5Throwingadietwice–samplemean假設只有上帝知道X的機率分配,平均數m是
4、3.5,變異數s2是2.92,而你不知道。你希望丟骰子兩次(樣本數n=2),藉由樣本平均點數估計平均數m。對於以樣本平均數來估計m,我們想重複抽樣很多次,甚至是抽出所有可能情形,看有怎麼樣的結果發生...AUCh9-6Throwingadietwice–samplemeanAUCh9-7Thedistributionofwhenn=211.52.02.53.03.54.04.55.05.56.06/365/364/363/362/361/36的平均數=1.0(1/36)+1.5(2/36)+….=3.5的變異數=(1.0-3.
5、5)2(1/36)+(1.5-3.5)2(2/36)...=1.46AUCh9-8SamplingDistributionoftheMean教科書第308頁表9.2教科書第308頁圖9.2如表9.2與圖9.2所示，樣本平均數也有自己的機率分配，在此將樣本平均數的機率分配改名為(樣本)平均數的抽樣分配。教科書第309頁第11-14行AUCh9-9SamplingDistributionoftheMeanAUCh9-10SamplingDistributionoftheMeanNoticethatissmallerthans2x.
6、Thelargerthesamplesizethesmaller.Therefore,tendstofallclosertom,asthesamplesizeincreases.抽樣分配的標準差改名為標準誤(standarderror)課本第309頁最後一段至311頁第一段AUCh9-11SamplingDistributionoftheMeanDemonstration:Thevarianceofthesamplemeanissmallerthanthevarianceofthepopulation.123Mean=1.5M
7、ean=2.5Mean=2.Population1.51.51.51.51.51.51.51.51.51.51.51.51.52.52.52.52.52.52.52.52.52.52.52.52.52.522222222222Comparethevariabilityofthepopulationtothevariabilityofthesamplemean.LetustakesamplesoftwoobservationsAUCh9-12Also,Expectedvalueofthepopulation=(1+2+3)/3=
8、2Expectedvalueofthesamplemean=(1.5+2+2.5)/3=2SamplingDistributionoftheMeanAUCh9-13SamplingDistributionoftheMeanIntermsofmeanandvariance,We
