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

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Stat 312: Lecture 14Large sample testMoo K. Chung[email protected]October 28, 20041. Testing meanμwith unknown varianceσ2.H0:μ=μ0vs.H1:μ < μ0Test statistic:t=¯x-μ0s/n.Rejection region forlevelαtest:z≤ -tα,n-1.Ex.IQ of a dog,XiN(μ, σ2),whereσisunknown. TestH0:μ= 100 vs.H1:μ <100at levelα= 0.05.> x<-c(30, 25, 70, 110, 40, 80, 50, 60,100, 60)> t=(mean(x)-100)/(sd(x)/sqrt(10))> t[1] -4.205156> qt(0.05,9)[1] -1.833113A simpler method is to use commandt.test.>help(t.test)...t.test(x,alternative=c("two.sided", "less","greater"),conf.level = 0.95)...> t.test(x,mu=100,alternative="less",conf.level=0.95)One Sample t-testdata:x t = -4.2036, df = 9,p-value = 0.001147alternative hypothesis: true mean isless than 10095 percent confidence interval:-Inf 78.8531sample estimates: mean of x 62.52. Testing on meanμwhen the sample size islarge.H0:μ=μ0vs.H1:μ < μ0Test statistic:z=¯x-μ0s/n.Rejection region forlevelαtest:z≤ -zα.3. Letpbe the proportion of a population with aspecified property. Assume the sample size to
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