hw9 - Math 3C Homework 9 Solutions Ilhwan Jo and Akemi...

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Unformatted text preview: Math 3C Homework 9 Solutions Ilhwan Jo and Akemi Kashiwada [email protected], [email protected] Assignment: Section 12.6 Problems 15, 16, 17(a), 18(a), 19(a), 20(a), 23-26 15. Toss a fair coin 400 times. Use the central limit theorem to find an approximation for the probability of at most 190 heads. Solution Let X i = ( 1 , if i th toss is heads , otherwise . Since the X i are i.i.d. and binomially distributed we know μ = EX i = np = (1) 1 2 ¶ = 1 2 , σ 2 = var( X i ) = np (1- p ) = (1) 1 2 ¶ 1- 1 2 ¶ = 1 4 . Now let S 400 = ∑ 400 i =1 X i count the number of heads we get out of 400 coin tosses. By the central limit theorem we get P ( S 400 ≤ 190) = P S 400- 400 μ √ 400 σ 2 ≤ 190- 200 10 ¶ = P S 400- 400 μ √ 400 σ 2 ≤ - 1 ¶ ≈ 1- Φ(1) = 1- . 8413 = 0 . 1587 16. Toss a fair coin 150 times. Use the central limit theorem to find an approximation for the probability that the number of heads is at least 70. Solution Let X i be defined as in the previous problem and let S 150 = ∑ 150 i =1 X i . By the central limit theorem P ( S 150 ≥ 70) = P S 150- 150 μ √ 150 σ 2 ≥ 70- 75 √ 37 . 5 ¶ = 1- P S 150- 150 μ √ 150 σ 2 ≤ - . 82 ¶ ≈ 1- Φ(- . 82) = Φ(0 . 82) = 0 . 7939 1 17a. Toss a fair coin 200 times. Use the central limit theorem to find an approximation for the probability that the number of heads is at least 120. Solution Let X i be defined above and let S 200 = ∑ 200 i =1 X i . By the central limit theorem P ( S 200 ≥ 120) = P S 200- 200 μ √ 200 σ 2 ≥ 120- 100 √ 50 ¶ = 1- P S 200- 200 μ √ 200 σ 2 ≤...
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hw9 - Math 3C Homework 9 Solutions Ilhwan Jo and Akemi...

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