Comparing two domain means in an SRS. Suppose there are two domains, defined by indicator variable

Then, letting ui = xiyi, the population values of the two domain means are

If an SRS of size n is taken from a population of size N, the population domain means may be estimated by

a. Use an argument similar to that in the discussion following (4.5) to show that

b. For an SRS, show using (A.10) that

[Consequently, since Property 7 of Expected Value in Section A.2 implies that V (1− 2) = V (1) + V (2) − 2Cov (1, 2), in an SRS V (1 − 2) ≈ V (1) + V (2) and an approximate 95% CI for U1 − U2 is given by

Thus, for an SRS, the large-sample CI for the difference of two domain means is the same (if we ignore the fpc) as you learned in your introductory statistics class. Note, though, that this result holds only for an SRS. For more complex sampling designs the covariance of the estimated domain means may be nonzero.