(Solution) - Suppose in a two stage cluster sample that all population cluster -(2025 Original AI-Free Solution)

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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 ? unb = ?r , 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