Midterm_I_Solutions - ISyE 2028 A Midterm I Friday A...

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ISyE 2028 A, Midterm I, Friday 6/6/2008, A. Megahed and J. Vastola 1 Midterm I Solutions ISyE 2028 A, Summer, 2008 June 6, 2008 3:20pm - 4:30pm Name: . GT ID #: . 1. This exam is a closed book and notes exam. You may only use the materials which we have provided at the beginning of the exam. 2. You may use a calculator. However, during the exam, it is not allowed to share a calculator. 3. There are five problems totalling 100 points. Show all work to receive full credit. 4. Please read and sign the honor code statement. Honor code statement. I pledge my honor that: I have completed all problems on the exam on my own, I have not used any unauthorized materials while completing and preparing for this exam, and I have not given anyone else access to my exam. Honor code signature: .
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ISyE 2028 A, Midterm I, Friday 6/6/2008, A. Megahed and J. Vastola 2 1. (20 points) Given a random sample X 1 , X 2 , . . . , X n from a normal distribution with mean 36 and variance 15, find the sample size n that is required so that P (34 . 8 ¯ X 37 . 2) = 0 . 90 . Solution. We are given μ = 36 and σ 2 = 15. Then by the Central Limit Theorem, we have P (34 . 8 ¯ X 37 . 2) = 0 . 90 P ( 34 . 8 - μ q σ 2 /n ¯ X - μ q σ 2 /n 37 . 2 - μ q σ 2 /n ) = 0 . 90 P ( 34 . 8 - 36 q 15 /n Z 37 . 2 - 36 q 15 /n ) = 0 . 90 P ( - 0 . 3098 n Z 0 . 3098 n ) = 0 . 90 We know that P ( - 1 . 645 Z 1 . 645) = 0 . 90 , thus we must solve . 3098 n = 1 . 645 n = 28 . 1948 .
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