Question

Question 1.
Consider a stratified design composed of H strata of size Nh, h = 1,...,H. We want to estimate the population mean µy of the characteristic y. Let µx,h, h =1,...,H be the means in the strata (in the population) of an auxiliary characteristic x. The µx,h are supposedly known and we propose to estimate µy using the following estimator: b µD = yst +µx −xst where yst and xst are the basic estimate of the population means µy and µx for y and x, respectively.

(a) Give an expression of µx in terms of µx,h, h =1,...,H.

(b) Show thatb µD is unbiased estimates for µy.

(c) Give the variance ofb µD

(d)Let n =PH h=1 nh be the sample size. What is the optimal allocation of the nh in order to minimise the variance ofb µD? We consider that the init cost of the survey does not depend on the stratum. (e) (1 mark) In which favourable case isb µD preferable to yst?

Answer 1

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nhala Na Sh . n = che rish (0) Com tradis using za t no i jelo ni Ni Shi de ть - |ть - N, s, а на 4. Inasan ③ Where th Where there is a strong correlations between x and y then var (lo) is minimum and Ms is preferable to Y. ther Name Cup es wie din isto