Suppose that a certain population of individuals is composed of k different strata (k ≥ 2), and that for i = 1, . . . , k, the proportion of individuals in the total population who belong to stratum i is pi, where pi > 0 and = 1. We are interested in estimating the mean value μ of a certain characteristic among the total population. Among the individuals in stratum i, this characteristic has mean μi and variance σ2i , where the value of μi is unknown and the value of σ2 i is known. Suppose that a stratified sample is taken from the population as follows: From each stratum i, a random sample of ni individuals is taken, and the characteristic is measured for each of these individuals. The samples from the k strata are taken independently of each other. Let Xi denote the average of the ni measurements in the sample from stratum i.
a. Show that μ = and show also that = is an unbiased estimator of μ.
b. Let denote the total number of observations in the k samples. For a fixed value of n, find the values of n1, . . . , nk for which the variance of will be a minimum.