During the Last Few Days Before a Presidential Election, There $\hat { p }$

Essay

During the last few days before a presidential election, there is a frenzy of voting intention surveys. On a given day, quite often there are conflicting results from three major polls.
(a)Think of each of these polls as reporting the fraction of successes (1s)of a Bernoulli random variable Y, where the probability of success is Pr(Y = 1)= p. Let $\hat { p }$ be the fraction of successes in the sample and assume that this estimator is normally distributed with a mean of p and a variance of $\frac { p ( 1 - p ) } { n }$ Why are the results for all polls different, even though they are taken on the same day?
(b)Given the estimator of the variance of $\hat { p }$ , $\frac { \hat { p } ( 1 - \hat { p } ) } { n }$ , construct a 95% confidence interval for $\hat { p }$ For which value of $\hat { p }$ is the standard deviation the largest? What value does it take in the case of a maximum $\hat { p }$ ?
(c)When the results from the polls are reported, you are told, typically in the small print, that the "margin of error" is plus or minus two percentage points. Using the approximation of 1.96 ? 2, and assuming, "conservatively," the maximum standard deviation derived in (b), what sample size is required to add and subtract ("margin of error")two percentage points from the point estimate?
(d)What sample size would you need to halve the margin of error?