Suppose in a two-stage cluster sample that all population cluster sizes are equal
(Mi = M for all i), and that all sample sizes for the clusters are equal (mi = m for all i).
a. Show (5.30).
b. Show that MSW = S2(1 ˆ’ R2a) and that

c. Using (a) and (b), express V () as a function of n, m, N, M, and R2a.
d. Show that if S2 and the sample and population sizes are fixed, and if (mˆ’1)/m > n/N, then V () is an increasing function of R2a.

N(M 1)R N 1 MSB = 5