Question 1 We are interested in estimating the population mean height Y of year 1 pupils in a district of N = 2000 year 1 pupils. We propose to select a simple random sample without replacement. Suppose that the distribution of heights has a coefficient of variation S/Y = 0.50 where S = (N-1)i (yi-). a) Give an analytic expression for the 95% confidence interval for the mean height of year 1 pupils. Define all symbols used. b) What minimum sample size would be required if we want the relative standard error (s.e(y)/Y) to be less than 5%? Here, s.e(y) denotes the standard error of the sample mean y. [2] [6] c) Suppose that we would like to estimate the mean height Y of year 1 male pupils. Define the estimator you would use. Is this estimator biased? [4] d) Let be the unbiased estimator of the total height of year 1 male pupils. The variance estimator of M is given by vr(M)= where n is the sample size and S is given. Give an analytic expression for an estimator for the variance of the estimator found in c). This estimator should be a function of the proportion of year 1 male pupils in the sample and of S/. [13]