Assume That Two Presidential Candidates, Call Them Bush and Gore $\hat { p }$

Assume that two presidential candidates, call them Bush and Gore, receive 50% of the votes in the population. You can model this situation as a Bernoulli trial, where Y is a random variable with success probability Pr(Y = 1)= p, and where Y = 1 if a person votes for Bush and Y = 0 otherwise. Furthermore, let $\hat { p }$ be the fraction of successes (1s)in a sample, which is distributed N(p, $\frac { p ( 1 - p ) } { n }$ )in reasonably large samples, say for n ? 40.
(a)Given your knowledge about the population, find the probability that in a random sample of 40, Bush would receive a share of 40% or less.
(b)How would this situation change with a random sample of 100?
(c)Given your answers in (a)and (b), would you be comfortable to predict what the voting intentions for the entire population are if you did not know p but had polled 10,000 individuals at random and calculated $\hat { p }$ ? Explain.
(d)This result seems to hold whether you poll 10,000 people at random in the Netherlands or the United States, where the former has a population of less than 20 million people, while the United States is 15 times as populous. Why does the population size not come into play?