week11.pdf - STA247H1F Week 11 Central Limit Theorem Fall...

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STA247H1FWeek 11: Central Limit TheoremFall 2014STA247H1FFall 20141 / 16
Outline1Sampling Distributions2The Distribution of the Sample Mean3Sampling Distribution of The Sample MeanSTA247H1FFall 20142 / 16
Sampling DistributionsSampling DistributionsAstatisticis any quantity that is calculated from sample data; forexample, the mean, the sample variance, the proportion, etc.The statistic varies from sample to sample and hence it is a randomvariable and has a probability distribution called thesampling distribu-tion.The knowledge of the sampling distribution of a statistic helps to makeinference about the correspondingpopulation (true) parameter.The random variablesX1, X2,· · ·, Xnare said to form a random sam-ple of sizenif1TheXi’s are independent random variables.2EveryXihas the same probability distribution.STA247H1FFall 20143 / 16
Sampling DistributionsDeveloping a Sampling DistributionAssume there is a population as follows:Population sizeN= 4Random variable,X,is age of individualsValues ofX: 18, 20, 22, 24 (years)STA247H1FFall 20144 / 16
Sampling DistributionsDeveloping a Sampling DistributionSummary Measures for the Population Distribution:Mean:μ=Ni=1xiN=18+20+22+244= 21standard deviation:σ=qNi=1(xi-μ)2N=5 = 2.236STA247H1FFall 20145 / 16
Sampling DistributionsDeveloping a Sampling DistributionNow consider all possible samples of sizen= 2STA247H1FFall 20146 / 16
Sampling DistributionsSampling Distribution of All Sample MeansSTA247H1FFall 20147 / 16

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