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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 %Best Essay Writing Services 24/7 | UniversityEssayServices.com% and a stratum sample mean %Best Essay Writing Services 24/7 | UniversityEssayServices.com% 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 %Best Essay Writing Services 24/7 | UniversityEssayServices.com% and %Best Essay Writing Services 24/7 | UniversityEssayServices.com% Let %Best Essay Writing Services 24/7 | UniversityEssayServices.com% and consider the two estimators
%Best Essay Writing Services 24/7 | UniversityEssayServices.com%
where we assume that %Best Essay Writing Services 24/7 | UniversityEssayServices.com%
(a) Compute the biases, variances. and MSEs of %Best Essay Writing Services 24/7 | UniversityEssayServices.com% How should %Best Essay Writing Services 24/7 | UniversityEssayServices.com% be chosen to make  %Best Essay Writing Services 24/7 | UniversityEssayServices.com% unbiased?
{b) Neyman allocation. Assume that %Best Essay Writing Services 24/7 | UniversityEssayServices.com% are known (estimates will be used in a later chapter). Show that the strata sample sizes that minimize %Best Essay Writing Services 24/7 | UniversityEssayServices.com% are given by
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(c) Show that %Best Essay Writing Services 24/7 | UniversityEssayServices.com% with %Best Essay Writing Services 24/7 | UniversityEssayServices.com% minus MSE %Best Essay Writing Services 24/7 | UniversityEssayServices.com% with nk given by (1.7.3) is
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