7.

(a) For m ≥ 2, let X1, X2, . . . , Xm be a random sample from a Bernoulli distribution with parameter p. Let ¯X = (X1 + X2 + · · · + Xm)/m. Show that Var( ¯X ) ≤ 1/(4m) ≤ 1/8.

(b) For a study, a sociologist needs to estimate p, the proportion of foreign-born residents of California. To do so, she takes n (n ≥ 2) random samples from the residents of California independently and measures p1, p2, . . . , pn, the proportions of foreign-born residents in those samples. Determine how large n should be so that, with a probability of at least 0.96, we have

Note that the sizes of the n samples taken need not be equal.

(c) Do part

(b) if the sizes of the n samples taken are all equal to 50.

P+ P++Pn n P