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sample.mid.sol

# sample.mid.sol - Y follows a binomial distribution with n =...

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Solutions to STAT 200 Sample Midterm Exam (Section 103) Problem 1. B, C, B, C, C, A, C, C, C, C. Problem 2. (a) P (50 < X < 80) = P ( 50 - 63 8 < Z < 80 - 63 8 ) = P ( - 1 . 625 < Z < 2 . 125) = 0 . 9304 . (b) We need to find the value of z so that P ( Z < z ) = 0 . 10. From Table A, we have z = - 1 . 28. So X - 63 8 = - 1 . 28 , X = 63 - 8 × 1 . 28 = 52 . 76 . Problem 3. (a) Let X be the number of successful guesses. We have n = 100 , p = 0 . 25. So μ X = np = 25 , σ X = np (1 - p ) = 4 . 33 . (b) Since np = 25 > 10 and n (1 - p ) = 75 > 10, we have ˆ p N (0 . 25 , 0 . 0433) approximately. Thus P p 0 . 30) = P ( Z 0 . 30 - 0 . 25 0 . 0433 ) = 0 . 8749 . (c) X Binomial(100 , 0 . 25). So P ( X 30) = P ( X = 0)+ P ( X = 1)+ · · · + P ( X = 30) = 30 k =0 100! k !(100 - k )! × 0 . 25 k × 0 . 75 100 - k . Problem 4. (a) Let X be the ACT score for a randomly selected student. Then X N (18 , 6). We have ¯ X = 1 100 100 i =1 X i N (18 , 6 10 ) = N (18 , 0 . 6) . Thus P (17 ¯ X 19) = P ( 17 - 18 0 . 6 Z 19 - 18 0 . 6 ) = P ( - 1 . 67 Z 1 . 67) = 0 . 9525 - 0 . 0475 = 0 . 90 . 1

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(b) (10 pts) Let p be the probability that a randomly selected student is eligible for the honour program. Then p = P ( X > 25) = P ( Z > 25 - 18 6 ) = P ( Z > 1 . 17) = 0 . 12 . Let Y be the number of eligible students among the 100 students.
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Unformatted text preview: Y follows a binomial distribution with n = 100 and p = 0 . 12, and E ( Y ) = np = 100 × . 12 , V ( Y ) = np (1-p ) = 100 × . 12 × (1-. 12) . Thus, based on the Central Limit Theorem, we have P ( Y ≥ 15) = P ( Y > 14 . 5) ≈ P ( Z > 14 . 5-100 × . 12 q 100 × . 12 × (1-. 12) ) = P ( Z > . 77) = 0 . 22 . Problem 5. (a) No. This is an observational study. Perhaps Anesthetic C is used for some more diﬃcult surgeries. (b) No mention is made of randomization of anesthetic for each patient. For an experiment, each patient would be equally likely to receive one of A, B, C, or D. (c) No. Some patients may be sensitive to some kinds of anesthetic. 2...
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sample.mid.sol - Y follows a binomial distribution with n =...

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