Comparing two domain means in a cluster sample. In Exercise 24 of Chapter 4, you showed that in an SRS where y1 and y2 estimate respective population domain means yU1 and yU2, V(y1 −y2) ≈ V(y1)+V(y2) because Cov (y1, y2) ≈ 0. Now let’s explore what happens when a one-stage cluster sample is selected from a population of N psus. For simplicity, assume that each psu has M ssus and that an SRS of n psus is selected. Let 1 and 2 be the estimators of the domain means from the cluster sample. Similarly to Exercise 24 of Chapter 4, let xij = 1 if ssu j of psu i is in domain 1 and xij = 0 if ssu j of psu i is in domain 2, and let uij = xijyij .
a Find

b. Show that the covariance in (a) is 0 if for each psu, all of the elements in that psu belong to the same domain—that is, either tix = 0 or tix = M for each psu i. [If the covariance in (a) is 0, then (4.26) implies that Cov (1, 2) ≈ 0.]
c. Give an example in which the covariance in (a) is not 0.