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STA 304H1F-1003H Fall 2019
Assignment 2-Question 4
Question 1. (5 marks)
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:
µbD = yst + µx − xst
where yst and xst are the basic estimate of the population means µy and µx for y and x,
(a) (1 mark) Give an expression of µx in terms of µx,h, h = 1, . . . , H.
(b) (1 mark) Show that µbD is unbiased estimates for µy.
(c) (1 mark) Give the variance of µbD
(d) (1 mark) Let n =PH
h=1 nh be the sample size. What is the optimal allocation of the nh
in order to minimise the variance of µbD? We consider that the init cost of the survey
does not depend on the stratum.
(e) (1 mark) In which favourable case is µbD preferable to yst?

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