统计学

1ChapterFiveParameterEstimate2MainpointsMethodsofParameterEstimatePointEstimationIntervalEstimationDeterminingtheSampleSize3populationknownHowaboutthesamplemeanXSampleknownXChapter4Chapter5HowaboutthePopulation4Section1PointandIntervalEstimation5Mainpoints•PointEstimation点估计•PropertiesofGoodPointEstimators–Unbiasedness无偏性–Efficiency有效性–Consistency一致性•BasicPrincipleofIntervalEstimate6PointEstimation：如果使用估计量的单一值作为总体参数的估计值，那么这种估计则为点估计。P249-250（矩估计法、极大似然估计法）估计量的评价标准：P271-2731.Unbiasedness：无偏性考虑的是估计量和参数的系统偏差问题2.Efficiency：评价无偏估计量的离中趋势即分散程度。设与是参数的无偏估计量，若，则称是比有效的估计量。3.Consistency：如果样本容量增大时，估计量可以以较大的概率接近所要估计的总体参数，则称为一致的估计量。对于,p,s2都满足无偏性、有效性和一致性。)ˆ(E)ˆ()ˆ(211ˆ2ˆ1ˆ2ˆX1ˆlimpn7BasicPrincipleofIntervalEstimate据上一章我们知道，如样本平均数服从正态分布那么有:也就是说有95.44%的样本平均数在之间,从这个可以推出:有95.44%的把握区间EquationpXXX2209544.22XX,XXXX2,2包含8ConfidenceIntervalsInterpretingtheConfidenceLevel•Theconfidencelevelistheprobabilitythattheprocedureusedtodeterminetheintervalwillprovideanintervalthatincludesthepopulationparameter.•Ifweconsiderallpossiblerandomlyselectedsamplesofsamesizefromapopulation,theconfidencelevelisthefractionorpercentofthosesamplesforwhichtheconfidenceintervalincludesthepopulationparameter.Note:Oftenexpresstheconfidencelevelasapercent.Commonlevelsare90%,95%and99%.Confidenceinterval:anintervalofvaluescomputedfromsampledatathatislikelytoincludethetruepopulationvalue.9Constructinga95%ConfidenceIntervalforaParameterPointestimatorMarginoferrorInthelongrun,about95%ofallconfidenceintervalscomputedinthiswaywillcapturethepopulationparameter,andabout5%ofthemwillmissit.Becareful:Theconfidencelevelonlyexpresseshowoftentheprocedureworksinthelongrun.Anyonespecificintervaleitherdoesordoesnotincludethetrueunknownpopulationvalue.10ConfidenceCoefficient置信系数andConfidenceInterval置信区间1-α的区间包含μα的区间不包括μCriticalValueoftheIntervalXzX2/XSamplingDistributionofα/21-αα/2XP29011Recall:•Aparameterisapopulationcharacteristic–valueisusuallyunknown.Weestimatetheparameterusingsampleinformation.•Astatistic,orestimate,isacharacteristicofasample.Astatisticestimatesaparameter.•Aconfidenceintervalisanintervalofvaluescomputedfromsampledatathatislikelytoincludethetruepopulationvalue.•Theconfidencelevelforanintervaldescribesourconfidenceintheprocedureweused.Weareconfidentthatmostoftheconfidenceintervalswecomputeusingaprocedurewillcontainthetruepopulationvalue.12区间估计的几个关键概念•置信系数1-α使人相信区间包含总体参数的概率,一般取0.95,0.90,0.99.它的大小说明估计的把握性的大小.•置信区间:在一定概率的保证下,包含总体均值的区间,区间的宽窄说明估计精度的大小.区间越宽,估计的精度就小;否则就大.•临界值:置信区间的上限和下限•注意置信系数和区间宽窄的关系P29113StandardErrorsRoughDefinition:Thestandarderrorofasamplestatisticmeasures,roughly,theaveragedifferencebetweenthestatisticandthepopulationparameter.This“averagedifference”isoverallpossiblerandomsamplesofagivensizethatcanbetakenfromthepopulation.TechnicalDefinition:Thestandarderrorofasamplestatisticistheestimatedstandarddeviationofthesamplingdistributionforthestatistic.14Poll:Classof175students.Inatypicalday,abouthowmuchtimetoyouspendwatchingtelevision?StandardErrorofaSampleMeanExample:MeanHoursWatchingTVdeviationstandardsample,).(.snsxesVariableNMeanMedianTrMeanStDevSEMeanTV1752.092.0001.9501.6440.124124.175644.1..nsxes15Poll:Randomsampleof935Americans“Doyouthinkthereisintelligentlifeonotherplanets?”StandardErrorofaSampleProportionExample:IntelligentLifeonOtherPlanetsproportionsample,1).(.pnpppesResults:60%ofthesamplesaid“yes”,=.60016.9356.16...pespThestandarderrorof.016isroughlytheaveragedifferencebetweenthestatistic,,andthepopulationparameter,π,forallpossiblerandomsamplesofn=935fromthispopulation.p16StepsofIntervalEstimate•DeterminingofConfidenceCoefficient•DeterminingoftheProbabilityofDistributionoftheStatistic•SamplingwiththeSampleSizen•Computing,andcomputingtheSifσisunknown•DeterminingtheCriticalValueoftheIntervalXXzXXtsnpzp17Section2PracticeofIntervalEstimateP30118•Normaldistributionwithσassumedknown•Anydistributionwithσassumedknown–Large-samplecase•IntervalEstimateofaPopulationMean:Large-SampleCasewithσEstimatedbys•IntervalEstimateofaNormalPopulationMean:Small-SampleCasewithσEstimatedbys•IntervalEstimateofaPopulationProportionXzXXzXpzpnszXXtsnMainpoints19Example1Normaldistributionwithσassumedknown某质量管理部门的负责人估计一批原材料的平均重量。抽取样本容量为250的一个随机样本，测得样本平均数为为65千克。已知总体标准差为15千克，假设原材料每包的重量服从正态分布。求置信系数为95%的这批原材料平均重量的置信区间。20解：根据已知条件可知样本平均数服从正态分布。的置信区间的临界值为651961525065186..ResultofExample1？21Example2Anydistributionwithσassumedknown:Large-samplecase某职业介绍所的职员从申请某一职业的1000名申请者中采用不重复抽样方式随机抽取了200名申请人，借此来估1000名申请者考试的平均成绩。样本平均数为78分，由已往经验得知总体的方差为90分。求总体平均数的90%的置信区间。22解：由于n大于30，根据中心极限定理服从正态分布XnNnN19020010002001000106.的置信区间为7816450678164506..,..得出：有90%的把握总体均值在（77,79）区间之内。XResultofExample223Example3IntervalEstimateofaPopulationMean:Large-SampleCasewithσEstimatedbys某百货商店通过100位顾客的随机样本研究购买额。样本均值为247.5元，样本标准差为55元。求逛此商店的所有顾客平均购买额的99%的置信区间。24解：总体标准差未知时，严格来说，样本X应服从t分布，但由于n=100大于30。所以可以用正态分布逼近它。XzSn24752585510247514192333126169.....,.的置信区间为ResultofExample325Exercises1•P29