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

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Stat 312: Lecture 6 Confidence Intervals II. Moo K. Chung [email protected] February 6, 2003 Concepts 1. Let X i N ( μ, σ 2 ) with known σ 2 and unknown μ . 100(1 - α )% confidence interval for μ is. ˆ μ L = ¯ x - z α/ 2 · σ/ n, ˆ μ U = ¯ x + z α/ 2 · σ/ n. 2. The sample size is inversely related to the width of confidence interval. 3. Central Limit Theorem. Let X 1 , · · · , X n be a ran- dom sample with mean μ and variance σ 2 . For large n , Z = ¯ X - μ σ/ n N (0 , 1) . 4. If n is sufficiently large, approximate 100(1 - α )% confidence interval for μ is ¯ x ± z α/ 2 s/ n , where s is the sample standard deviation. In-class problems Continuing Exercise 6.25, construct 98% confidence in- terval. Example 7.4. Response time N ( μ, σ 2 ) , σ = 25 . Find the sample size n that ensures 95 % CI with a width of 10. Example 7.6. Alternating current (AC) voltage data > data(xmp07.06) > attach(xmp07.06) > str(xmp07.06) ‘data.frame’:48 obs. of 1 variable: $ C1: int 62 50 53 57 ... > boxplot(C1) > mean(C1) [1] 54.70833
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Unformatted text preview: > sd(C1) 40 45 50 5 60 65 [1] 5.230672 >mean(C1)-qnorm(0.975)*sd(C1)/sqrt(length(C1)) [1] 53.2286 >mean(C1)+qnorm(0.975)*sd(C1)/sqrt(length(C1)) [1] 56.18807 Ex. Toss n = 100 biased coins with P ( H ) = p . Sup-pose you observe 38 heads. Construct 95 % CI of p . > X<-rbinom(100,1,0.4) > X [1] 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 [17] 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 [33] 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 [49] 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 [65] 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 [81] 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 1 [97] 0 1 1 0 > sd(X) [1] 0.4878317 > sqrt(0.38*(1-0.38)/(100-1)) [1] 0.04878317 > 0.38+1.96*0.049/sqrt(100) [1] 0.389604 > 0.38-1.96*0.049/sqrt(100) [1] 0.370396 Self-study problems Example 7.8., Exercise 7.13., 7.19., 7.25. In the above coin tossing example, check if ¯ X is MLE....
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