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

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Stat 312: Lecture 11 Testing on Population Means Moo K. Chung [email protected] February 25, 2003 Concepts 1. P (Type I error) = P (reject H 0 | H 0 true) = α. The resulting α is called the significance level of the test and the corresponding test is called a level α test . We will use test procedures that give α less than a specified level (0.05 or 0.01 usually). 2. Testing mean μ with known variance σ 2 . H 0 : μ = μ 0 vs. H 1 : μ < μ 0 Test statistic: z = ¯ x - μ 0 σ/ n . Rejection region for level α test: z ≤ - z α . 3. 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 . 4. Testing 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 . In-class problems Assume the IQ of a dog follows X i N ( μ, 10 2 ). The IQ of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We want to test if dogs are as smart as people by testing H 0 : μ = 100 vs. H 1 :
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Unformatted text preview: at level α = 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 > qnorm(0.05) [1] -1.644854 Assume the IQ of a dog follows X i ∼ N ( μ, σ 2 ) , where σ is unknown. Test H : μ = 100 vs. H 1 : μ < 100 at level α = 0 . 05. > t=(mean(x)-100)/(sd(x)/sqrt(10)) > t [1] -4.205156 > qt(0.05,9) [1] -1.833113 > 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 Self-study problems Example 8.6., 8.8., 8.9. Assignment III. Due March 13. 11:00am. 8.24., 8.26, 8.28., 8.36. (a) and (b) only, 8.38....
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