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
