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 that ṫ unb = ṫr , and, hence, that unb = r .
b. Fill in the formulas for the sums of squares in the ANOVA table below, for the sample data.

c. Show that E[msw] = MSW and

Where MSB and MSW are the between and within mean squares, respectively, from the population ANOVA table given in Table 5.1.
d. Show that

is an unbiased estimator of MSB.
e. Show, using (5.24) or (5.28), that

Source df Sum of Squares Mean Square Between psus I Within psusn-I) Total msb msw msto In Elmsbl MMSB+(1-MMSW, MSB msb n msb 1 msw Ni nmI