fin04sol - Suggested solution Question 1 Answer For 99...

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6 Suggested solution Question 1 Answer: For 99% certainty, z(.995)=2.575. (a) Since P can be any number between 0 and 1, P(1-P) maxizes at 0.5x0.5=0.25. n=(2.575/0.02)2 P(1-P)=(2.575/0.02)2 0.5(1-0.5)=16576.56x0.25=4144.14=4145 (b) Since P is between 0 and 0.1, P(1-P) maximizes at 0.1(1-0.1)=0.09 n=16576.56x0.09=1491.9=1492 Question 2 Question 3: Note that H 0 : p = 0.5 vs. H 1 : p < 0.5 where p is the probability of getting heads. (a) Let X be the number of heads in 10 tossings. Then, the probability of Type I error is given by Pr{X < 1} under H 0 . That is, X ~ Bin(10, 0.5). Therefore, Pr{X < 1} = Pr{X = 0} + Pr{X = 1} = 0.000977 + 0.009766 = 0.010743 (b) If CLT holds for n = 10, then ˆ p = X/10 ~ N(0.5, 0.5(0.5)/10) under H 0 . In that case, the probability of Type I error is given by: Pr{X < 1} = Pr{ ˆ p < 0.1} = Pr{Z < (0.1 0.5) / 0.5(0.5)/10 } = Pr{Z < 2.53} = 0.0057 (c) The discrepancy is due to the fact that n = 10 is too small for CLT. Therefore, 0.010743 is more close to the true probability of Type I error.
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fin04sol - Suggested solution Question 1 Answer For 99...

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