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Suppose there is a first phase sample s 1 of m units, which is a simple random sample without replacement (SRSWOR) from a population of size N . Membership of a subpopulation of interest (referred to as A) is collected from this rst phase sample. Let N A be the population size of the subpopulation and N B be the remaining population. Write a i equal to 1 for the subpopulation and 0 otherwise. The first phase survey also collects auxiliary variables z i which may include a i . Let m A and m B be the first phase sample sizes of subpopulation members and non-members respectively.