Math 3C Homework 9 Solutions

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1 Math 3C Homewk 9 s Ilhwan Jo and Akemi Kashiwada Assignment: Section 1.6 Problems 15, 16, 17a, 18a, 19a, 0a, Toss a fair coin 400 times. Use the central limit theem to find an approximation f the probability of at most 190 heads. 1, if ith toss is heads. 0, otherwise Since the X i are i.i.d. and binomially distributed we know 1 µ = E np = 1 = 1 1, σ = varx i = np1 p = = 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 theem we get S µ P S = P 400σ 10 S µ = P 1 400σ 1 1 = = Toss a fair coin 150 times. Use the central limit theem to find an approximation f the probability that the number of heads is at least 70. X i be defined as in the previous problem and let S 150 = 150 i=1 X i. By the central limit theem S µ P S = P 150σ 37.5 = 1 P S µ 150σ = 0.8 =

2 17a. Toss a fair coin 00 times. Use the central limit theem to find an approximation f the probability that the number of heads is at least 10. X i be defined above and let S 00 = 00 i=1 X i. By the central limit theem S00 00µ P S = P 00σ 50 = 1 P 1.83 = = S00 00µ 00σ.83 18a. Toss a fair coin 300 times. Use the central limit theem to find an approximation f the probability that the number of heads is between 140 and 160. X i be defined above and let S 300 = 300 i=1 X i. By the central limit theem [ ] S µ P S 300 [140, 160] = P, = P S µ = = P S µ a. Suppose S n is binomially distributed with parameters n = 00 and p = 0.3. Use the central limit theem to find an approximation f P 99 S n 101 without the histogram crection. Since S n is binomially distributed, S n = n i=1 X i where 1 if the k-th trial is successful X k = We have µ = E 1 p p = p = 0.3, σ = varx i = EX i EX i = 1 p p p = p1 p = = 0.1. By the central limit theem, P 99 S n 101 = P S n nµ = P 6.0 S n nµ = a. Suppose S n is binomially distributed with parameters n = 150 and p = 0.4. Use the central limit theem to find an approximation f P S n = 60 without the histogram crection.

3 Since S n is binomially distributed, we know By the central limit theem, µ = E p = 0.4, σ = varx i = p1 p = = 0.4. P S n = 60 = P = P Sn nµ Sn nµ P Z = 0 = 0 = = 0 where Z is the standard nmally distributed random variable How often should you toss a coin to be at least 90% certain that your estimate of P heads is within 0.1 of its true value? 1 if ith toss is heads Then X n = 1 n n i=1 X i is an estimate of the proption of tosses resulting in heads. If we let S n = n i=1 X i, then n Xn p = Sn nµ p1 p is approximately standard nmally distributed with µ = p = EX i, σ = varx i = p1 p. We want to find n so that P X n p , n 0.1 P n X n p 0.1 n 0.9. p1 p p1 p p1 p Since n X n p is approximately standard nmally distributed, p1 p n 0.1 p1 p n 0.1 p1 p 0.95 n 0.1 p1 p 1.65 n 16.5 p1 p where p1 p attains its maximum 1 4 when p = 1. This gives us n 68.06, which implies n = 69 would be sufficient.

4 4. How often should you toss a fair coin to be at least 90% certain that your estimate of P heads is within 0.01 of its true value? This question is basically same as 3. n p1 p n 0.01 p1 p 1.65 n 165 p1 p where p1 p attains its maximum 1 4 when p = 1. This gives us n , which implies n = 6807 would be sufficient. 5. To fecast the outcome of a presidential election in which two candidates run f office, a telephone poll is conducted. How many people should be surveyed to be at least 95% sure that the estimate is within 0.05 of the true value? We assume that each individual votes f one and only one of the two candidates. A be one of the two candidates and 1 if ith individual voted f A We assume that X i s are i.i.d. Then X n = 1 n n i=1 X i is an estimate of the proption of individuals who voted f A. If we let S n = n i=1 X i, then n X n p = S n nµ p1 p is approximately standard nmally distributed with µ = p = EX i, σ = varx i = p1 p. We want to find n so that As in previous questions, we get P X n p , n 0.05 p1 p n 0.05 p1 p n 0.05 p1 p 1.96 n = 384., which implies n = 385 would be sufficient. 6. A medical study is conducted to estimate the proption of people suffering from seasonal affected disder. How many people should be surveyed to be at least 99% sure that the estimate is within 0.0 of the true value? 1 if ith individual is suffering from the disder

5 We assume that X i s are i.i.d. Then X n = 1 n n i=1 X i is an estimate of the proption of individuals who is suffering from the disder. If we let S n = n i=1 X i, then n Xn p = Sn nµ p1 p is approximately standard nmally distributed with µ = p = EX i, σ = varx i = p1 p. We want to find n so that P X n p , Again, we get n 0.0 p1 p n 0.0 p1 p n 0.0 p1 p.58 n = , which implies n = 4161 would be sufficient.

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