Suppose there is a very tight political race and we want to gauge support for candidate A. Let Mn = 21-1X; denote the proportion of polled people supporting candidate A and let p denote the true proportion. Further suppose we want to poll enough people to ensure that we are within +1% of p 95% of the time. Stated mathematically, we would like to find n such that: P(Mn - p > 0.01) < 0.05 (1) when E(Xi) = p. (a) Show that the variance of Bernoulli(p) is maximized when p=0.5.
(b) Using the worst-case variance from part a) and a Normal approximation, find the n such that (1) is satisfied.
(c) Again using the worst-case variance, use Chebychev's inequality to find a bound for n.