Effect of weighting class adjustment on variances. Suppose that an SRS of size n is taken. Let Zi =1 if unit i is included in the sample and 0 otherwise, with P (Zi = 1) = n/N. Two weighting classes are used to adjust for non response; define xi =1 if unit I is in class 1 and 0 if unit i is in class 2. Let Ri =1 if unit i responds to the survey and 0 otherwise. Assume that the Ri’s are independent Bernoulli random variables with P (Ri =1) =xiφ1 + (1 − xi) φ2, and that Ri is independent of Z1. . . ZN. The sample sizes in the two classes are n1 = ƩNi =1 Zixi and n2 =ƩNi =1 Zi (1−xi); note that n1 and n2 are random variables. Similarly, the number of respondents in the two classes are n1R =ƩNi =1 ZiRixi and n2R = ƩNi =1 ZiRi (1 − xi). Assume the number of respondents in each group is sufficiently large so that E [nc / ncR] ≈ 1/φc for c = 1, 2. With these assumptions, the weighting class adjusted estimator of the mean,

Is approximately unbiased for the population mean yU (see Exercise 17). Find the approximate variance of ŷwc. Use Property A.4 of Conditional Expectation in
Section A.4.