A news agency wants to poll the population of registered voters in the United States (over 200,000,000) to find out how many would vote for the Republican candidate, Henry Washington, if the election were held today. They intend to take a random sample, with replacement, of 1,000 registered U.S. voters, record their preferences for \(\left(y_{i}=1ight)\) or against \(\left(y_{i}=0ight) M r\). Washington, and then use the 1,000 sampled outcomes to estimate the proportion of registered voters in favor of the candidate. They conduct the random sampling, and observe that \(\sum_{i=1}^{n} y_{i}=593\). Given their random sampling design, they are assuming that \(y_{i}^{\prime} s \sim\) iid \(\operatorname{Bernoulli}(p)\), where \(p\) is the proportion of registered voters in favor of Mr. Washington. They intend to use the sample mean, \(\bar{X}\), as an estimator for \(p\).

(a) What is the expected value of the estimator?

(b) What is the standard deviation of the estimator?

(c) Provide a lower bound to the probability that the estimate is within \(\pm .03\) of the true proportion of voters in favor of the candidate.

(d) What size of random sample would the agency need to use in order to generate an estimate, based on the sample mean, that would be within \(\pm .01\) of the true proportion of voters?

(e) What is the estimate of the proportion of voters in favor of the candidate?

(f) Would there be much gain, in the way of lower variance of the estimator, if the agency would have sampled without replacement instead of with replacement? Explain.