2. Eddy and Harrison play a game as follows. Eddy selects a real number 6 at random and writes it down on a piece of paper. Harrison can ask Eddy what 6 is. Each time Harrison asks, Eddy tosses a biased coin secretly and tells Harrison the number 6 k if a head comes up (with probability p = 0.3) and the number 6' + k if a tail comes up (with probability 1 p = 0.7) for some k > 0. Harrison knows the values of p and k, and has to estimate the true value of 0 after asking Eddy twice based on the numbers X1 and X2. (a) Dene Y = (X1 + X2) / 2. Construct an unbiased estimator of 6, namely T, based on Y. (b) Dene W _ Y, for X1 75 X2, _ X1 +0., for X1 = X2. Find the value of a such that W is an unbiased estimator of 6. (c) Intuitively, which estimator do you think is a better estimator, T or W? 6. The manufacturer of a sports car wants to estimate p, the proportion of people in a given income bracket who will be interested in a particular new model, by constructing a confidence interval. (a) With the same confidence level, which confidence interval do you prefer, a wider one or a narrower one? Why? (b) Suppose the company wants to obtain a narrow interval with 99% confidence level. What is the minimum required sample size which will assure the width of the confidence interval to be smaller than 0.1