大二上學(xué)期a概率與統(tǒng)計lecture

Lecture18RecapConfidenceIntervals統(tǒng)計分析的任務(wù)通過樣本的統(tǒng)計量來了解總體的參數(shù)?？傮w參數(shù)p樣本統(tǒng)計量總體：所要調(diào)查觀察的全部個體?？傮w個數(shù)用N表示。有限和無限。樣本：從總體中抽取出來調(diào)查觀察的個體。抽樣容量用n

表示。

ExampleWeareinterestedinthepopulationmean.Wewanttousethesampleobservationstoestimatethepopulationmean.5點估計對于數(shù)量型數(shù)據(jù)，例如，我們可以用對于0-1型數(shù)據(jù)，例如，我們可以用

樣本平均值的分布，即它的抽樣分布。7抽取的樣本不同，那么算出的平均值也不同。那么樣本的平均值會和總體的平均值（要找的值）差距很大嗎？

經(jīng)典的理論結(jié)果

Formofintervalestimate：(Pointestimate-,Pointestimate+)

(r1*Pointestimate,r2*Pointestimate)Canyouprovideareasonablerangefortheparameters?Pointestimate=Populationparameter?

ConfidenceIntervalsConfidenceIntervalsforanArbitraryParameterqSupposeX1,…,Xnformarandomsamplefromadistributioninvolvinganunknownparameterq.SupposethattwostatisticsA(X1,…,Xn)andB(X1,…,Xn)canbefoundsuchthat,regardlessofthetruevalueofq,

wheregisafixedprobability.IftheobservedvaluesofA(X1,…,Xn)andB(X1,…,Xn)areaandb,thenwesaythattheinterval(a,b)isaconfidenceintervalforqwithconfidencecoefficientg,orqliesintheinterval(a,b)withconfidenceg.ConfidenceIntervalsfortheMeanofaNormalDistributionWhentheVarianceisKnownSupposethattheobservedvaluesx1,…,xncomefromarandomsamplefromanormaldistributionwithunknownmeanmandknownvarianceConfidenceIntervalsfortheMeanofaNormalDistributionWhentheVarianceisKnownSotheconfidenceintervalformisExampleContinued:

95%ConfidenceIntervalformWhenisknownItiscalculatedthatTheconfidenceintervalformisLetConsiderintervalshavingtheform

95%ofallvaluesinsidepurpleband,coversinsidepurpleband,coversoutsidepurpleband,missesInterpretationofConfidenceIntervalsBeforeobtainingtheobservedvaluesoroverrepeatedsampling,thereisprobability95%thatthesamplemeanwilllieinthepurpleband,so95%ofallintervalsformedbywillcoverthepopulationmean,orWedonotstatethatmliesintheconfidenceintervalwithprobability95%.Wecanonlysaythatweare95%confident.ConfidenceIntervalsfortheMeanofaNormalDistributionSupposethatX1,…,Xn

formarandomsamplefromanormaldistributionwithunknownmeanmandunknownvariance.:theaupper-quantileoftn-1distribution.Usuallytheconfidencecoefficientg=0.9,0.95,0.99,etc.Leta=1-g.TheconfidenceintervalformisExampleContinuedItiscalculatedthat

Sothe90%confidenceintervalestimateofthepopulationmeanisConfidenceIntervalsfor

theVarianceofaNormalDistributionSupposethattheobservedvaluesx1,…,xncomefromarandomsamplefromanormaldistributionwithunknownmeanmandunknownvariance

ConfidenceIntervalsfor

theVarianceofaNormalDistribution

SotheconfidenceintervalforisSuppose