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Mathematics, 14.04.2020 23:32 adrian08022

The voters in California are voting for a new governor. Of the voters in California, a proportion p will vote for Frank, and a proportion 1 − p will vote for Tony. In an election poll, a number of voters are asked for whom they will vote. Let Xi be the indicator random variable for the event "the ith person interviewed will vote for Frank."1 A model for the election poll is that the people to be interviewed are selected in such a way that the indicator random variables X1, X2, ... are independent and have a Ber(p) distribution. (a) Suppose we use X¯ n to predict p. According to Chebyshev’s inequality, how large should n be (how many people should be interviewed) such that the probability that X¯ n is within 0.2 of the "true" p is at least 0.9? (Hint. solve this first for p = 1/2, and use the fact that p(1 − p) ≤ 1/4 for all 0 ≤ p ≤ 1.) (b) Answer the same question, but now X¯ n should be within 0.1 of p. (c) Answer the question from part (a), but now the probability should be at least 0.95. (d) If p > 1/2, then Frank wins; if X¯ n > 1/2, you predict that Frank will win. Find an n (as small as you can) such that the probability that you predict correctly is at least 0.9, if in fact p = 0.6. 7. Let X1, X2, ..., X144 be independent and identical

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