Could someone please help me with this question:
Consider the population of all people resident in a town. Let p be the proportion of these who have ever been prescribed glasses or contact lenses. An optician wants to estimate p, so she chooses a random sample of 512 residents and asks each one if they have ever been prescribed glasses or contact lenses. Let X be the number who answer “yes”.
(a) State the distribution of the random variable X. Hence write down the expectation and variance of X.
(b) Find P(X ≤ 200) if p = 1/3.
(c) Let Y = X/512. Show that Y is an unbiased estimator of p.
(d) Find the mean squared error of Y as a function of p.