lecture14 notes - Stat 312 Lecture 14 Large sample test Moo K Chung [email protected] 1 Testing mean with unknown variance 2 H0 = 0 vs H1 < 0 0 Test

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Stat 312: Lecture 14 Large sample test Moo K. Chung [email protected] October 28, 2004 1. Testing mean μ with unknown variance σ 2 . H 0 : μ = μ 0 vs. H 1 : μ < μ 0 Test statistic: t = ¯ x - μ 0 s/ n . Rejection region for level α test: z ≤ - t α,n - 1 . Ex. IQ of a dog, X i N ( μ, σ 2 ) , where σ is unknown. Test H 0 : μ = 100 vs. H 1 : μ < 100 at level α = 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] -1.833113 A simpler method is to use command t.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-test data: x t = -4.2036, df = 9, p-value = 0.001147 alternative hypothesis: true mean is less than 100 95 percent confidence interval: -Inf 78.8531 sample estimates: mean of x 62.5 2. Testing on mean μ when the sample size is large. H 0 : μ = μ 0 vs. H 1 : μ < μ 0 Test statistic: z = ¯ x - μ 0 s/ n . Rejection region for level α test: z ≤ - z α . 3. Let p be the proportion of a population with a specified property. Assume the sample size to