Stratified sampling. We want to estimate the mean J1. = E(X) of a population that has been divided (stratified) into s mutually exclusive parts (strata) (e.g., geographic locations or age groups). Within the jth stratum we have a sample of i.i.d. random variables  and a stratum sample mean  We assume that the s samples from different strata are independent. Suppose that the jth stratum has 100pj% of the population and that the jth stratum population mean and variances are  and  Let  and consider the two estimators

where we assume that 
(a) Compute the biases, variances. and MSEs of  How should  be chosen to make   unbiased?
{b) Neyman allocation. Assume that  are known (estimates will be used in a later chapter). Show that the strata sample sizes that minimize  are given by

(c) Show that  with  minus MSE  with nk given by (1.7.3) is


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