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

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Unformatted text preview: Stat 312: Lecture 08 Large sample confidence intervals Moo K. Chung mchung@stat.wisc.edu September 27, 2004 1. The sample size is inversely related to the width of confidence interval. Example 7.4. 2. Central Limit Theorem. Let X1 , · · · , Xn be a random sample with mean µ and variance σ 2 . For sufficiently large n, Z= ¯ X −µ √ ∼ N (0, 1). σ/ n 3. Let X1 , · · · , Xn be a random sample with mean µ. For sufficiently large n, Z= ¯ X −µ √ ∼ N (0, 1) S/ n [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 > sqrt(0.38*(1-0.38)/100)*1.96 [1] 0.09513574 > 0.38+0.095 [1] 0.475 > 0.38-0.095 [1] 0.285 5. One-sided confidence interval: An 100(1 − α)% upper confidence bound for θ is θ < x + zα ¯ ˆ Vθ where S is the sample standard deviation. If n is sufficiently large, approximate 100(1 − α)% confidence interval for µ is s x ± zα/2 √ , ¯ n where s is the sample standard deviation. 4. General large sample confidence interval. Supˆ pose θ is an unbiased estimator of some parameter θ, Then 100(1 − α)% confidence interval is ˆ θ + zα/2 ˆ Vθ. and a lower confidence bound for µ is θ > x − zα ¯ ˆ Vθ. Review Problems. Example 7.8, 7.10. ˆ In many applications, Vθ is a function of θ which makes computation of CI complicated. In this sitˆ uation, we need to estimate Vθ further. Example. Toss n = 100 biased coins with P (H) = p. Suppose 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 ...
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This note was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Fall '04 term at Wisconsin.

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