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

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Stat 312: Lecture 5 Confidence Intervals Moo K. Chung [email protected] February 3, 2003 Concepts 1. For population parameter θ , 95 % confidence inter- val ( ˆ θ L , ˆ θ U ) of θ is computed by solving P ( ˆ θ L θ ˆ θ U ) = 0 . 95 2. Let X i N ( μ, σ 2 ) with known σ 2 and unknown μ . 95 % confidence interval for μ is ˆ μ L = ¯ x - 1 . 96 · σ/ n, ˆ μ U = ¯ x + 1 . 96 · σ/ n. 3. 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. In-class problems Continuing Exercise 6.25. Assuming σ = 18 . 86 , find 95% confidence interval of μ . > b<-qnorm(0.975,0,sigma/sqrt(10)) > a<-qnorm(0.025,0,sigma/sqrt(10)) > b [1] 11.68933 > a [1] -11.68933 > mean(X)-b [1] 372.7107 > mean(X)-a [1] 396.0893 > qnorm(0.995) [1] 2.575829 > qnorm(0.99) [1] 2.326348 > qnorm(0.975) [1] 1.959964 > qnorm(0.95) [1] 1.644854 Self-study problems Example 7.2.,7.3.,7.4.,7.5.
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Unformatted text preview: Due Feb 13. 11:00am. Exercise 6.30., 7.4., 7.10., 7.14., 7.22., 7.26. Bonus Problem (Hard). Let X 1 , ··· , X n be a ran-dom sample with mean μ and variance σ 2 . Among all estimators of the form ˆ μ = ∑ n j =1 c j X J , find MVUE and prove that it is in fact MVUE. Hand in this prob-lem directly to the instructor during the office hour . No partial credit for this problem. Midterms One page note and a calculator are allowed. Midterm I. March 4, 11:00am-12:15pm (Tuesday). Midterm II. April 15, 11:00am-12:15pm (Tuesday). 4 problems will be given: 1 easy, 2 medium diffi-culty and 1 challenging problems. Mock midterm prob-lems will be posted on the WEB one week before the midterms....
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