# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 13Testing on Population MeansMoo K. Chung[email protected]October 26, 20041. Since we can minimize both type I and typeII errors simultaneously in constructing a testprocedure, we will control for fixed type I error(levelα) and make type II error as small aspossible. The corresponding test procedure iscalled alevelαtest.2. Testing meanμwith known varianceσ2.H0:μ=μ0vs.H1:μ < μ0Test statistic:z=¯x-μ0σ/n, which is a pivot underH0. Apivotis a statistic whose distribution isindependent of population paramters.Rejec-tion region for levelαtest:z≤ -zα.Ex. Intelligence quotient (IQ) is a number usedto express the relative intelligence of a person.An average person has the IQ of 100. AssumeIQ of a dog followsXiN(μ,102). The IQs of10 dogs are measured: 30, 25, 70, 110, 40, 80,50, 60, 100, 60. We want to test if dogs are assmart as people by testingH0:μ= 100 vs.H1:μ <100at levelα= 0.05.> x<-c(30, 25, 70, 110, 40, 80, 50, 60,100, 60)> mean(x)[1] 62.5> z<-(mean(x)-100)/(10/sqrt(10))> z[1] -11.85854
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