Covariance of estimated population totals in a cluster sample. Suppose a one-stage cluster sample is taken from a population of N psus, with inclusion probabilities πi.
Let tx and ty be the population totals for response variables x and y, and let tix and tiy be the totals of variables x and y in psu i.
a. Show that

b. If an SRS of n of the N psus is selected, with Ï€i = n/N and Ï€ik = (n/N) [ (nˆ’1) / (N ˆ’ 1)], show using part (a) that

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tix ty 1-πί Cov(t, f N2 txly /t Cov (tx, ty) = W-Th