(b) Show that conditional on the observed data, the variance of y∗ is ms2 R(1 − r−1)∕n2 and that the expectation of s2

∗ is s2 R(1 − r−1)(1 +

rn−1(n − 1)−1).

(c) Show that conditional on the sample sizes n and r (and the population Y-values), the variance of y∗ is the variance of yR times

(1 + (r−1)n−1(1−r/n)(1−r/N)−1), and show that this is greater than the expectation of U∗ = s2

∗(n−1 − N−1).