Assessing the Unusualness of a Sample Mean Using the Central Limit Theorem

An analysis using the Central Limit Theorem to assess the unusualness of a sample mean in statistical hypothesis testing.

Caleb Patterson

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AssessingtheUnusualnessofaSampleMeanUsingtheCentralLimitTheoremAssumethatapopulationisnormallydistributedwithameanof100andastandarddeviationof15.Woulditbeunusualforthemeanofasampleof3tobe115ormore?Whyorwhynot?Solution:LetXbethesamplemeanofsize3.Giventhepopulationmean#=100andstandarddeviationo=15.BytheCentralLimitTheorem,Xxfollowsanormaldistributionwithmeanx;=100andstandarddeviationofoz=F=1%=8.66.So,theprobabilityofgettingsamplemeantobe115ormoreisP(X2115)=1-P(X<115)LetZz=21%ThenZfollowsthestandardnormaldistributionN(O,1).So,

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