# unit7 - STA247 Unit 7 Central Limit Theorem Fall 2015...

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STA247Unit 7: Central Limit TheoremFall 2015STA247Fall 20151 / 11
Outline1Sampling Distributions2Sampling Distribution of The Sample MeanSTA247Fall 20152 / 11
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 arandom sam-pleof sizenif1TheXi’s are independent random variables.2EveryXihas the same probability distribution.STA247Fall 20153 / 11
Sampling DistributionsThe Distribution of the Sample MeanProperties of Sample mean and TotalLetX1, X2,· · ·, Xnbe a random sample from a distribution with meanμand standard deviationσ. LetX=Pni=1Xinbe the mean. Then1E(X) =μX=μ.2V(X) =σ2X=σ2nandσX=σpn(the standard error of the mean).Note that according to Result 1, the sampling distribution ofXis centeredprecisely at the mean of the population from which the sample has beenselected. Result 2 shows that theXdistribution becomes more concentratedaboutμas the sample sizenincreases.