Section101PointandIntervalEstimation101节点估计和区间估计

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1、PointandIntervalEstimationwithConfidenceAnotherSpectacularPowerPointPresentationPointEstimatesDefinition:Asinglenumberthatisbasedonsampledata,andrepresentsareasonablevalueofthecharacteristicofinterest.ThingsLike:xsM-Thesimplestapproachtoestimatingapopu

2、lationvalue.LetsLookatanExampleArecentsurveyaskedhighschoolseniorshowmanyhourstheyspentonlineeachday.Findthepointestimateofµwiththedatabelow.47.2555.256.256.254.53.54.25Whatcouldweusetoestimateµ?MorxFindboth:x=46.25=4.62510M=4.75Whichoneisbetter?Recallw

3、henitisbesttouseeachone…ConfidenceIntervals(estimatingµ)ThingstoRememberAbouttheDistributionofx:1)Hasanormaldistribution(byCLT)2)Isanunbiasedestimatorofµ(x=µ)3)Ifpermissionisgranted:s=σ/√nIssueswithPointEstimates:ifwere–dothesample,willwegetsamestat?Con

4、fidenceIntervalsDef:theintervalofreasonablevaluesofthecharacteristicofinterestPatternfortheformula:(what’sonAPtest)statistic±criticalvalue(standarddeviationofstat)ForEstimatingµ:x±z*(σ/√n)MarginofErrorConfidenceLevelsDef:thesuccessrateofthemethodusedtoc

5、onstructtheinterval.Ex:A95%confidenceinterval=ifwetooklotsofsamplesandfoundthemeanoverandoveragain,inthelongrun95%oftheresultingintervalswouldcapturethetruevalueoftheparameter.Ifwegothroughthesameprocesswiththesamesamplesize,σ,andz*,butdifferentx,95%oft

6、hoseintervalswillcontainµ.(CIapplet)CalculatingConfidenceIntervalsWhatisz*?Alocationonthenormalcurvethatresultsintheneededconfidencelevel.Ex:UseTableC!z*=1.96-z*z*95%CalculatingConfidenceIntervalsLet’stryanotherz*:Whatisz*?z*=2.326-z*z*98%CriticalValues

7、z*isalsocalledacriticalvalue.Anyideaswhy?TheareaoutsideoftheintervalarecalledcriticalareasPANIC(notatthedisco…)Whencalculatingaconfidenceinterval,remembertoPANIC.P=Parameterofinterest(whatyouareestimating).A=Assumptions(like10n…)N=Nametheinterval(x±z*·σ

8、/√n)I=Interval:lowtohighC=ConclusionincontextConclusionincontext:MantraIam95%confidentthatthetruepopulationmeanofcontextofproblemiscontainedinthisinterval[,](confidenceinterval).Ifthissameprocedurewererepeatedmanytimes,thenapprox