11. Of n1 randomly selected male smokers, X1 smoked lter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked lter cigarettes. Let p1 and p2 denote the probabilities that a randomly selected male and female, respectively, smoke lter cigarettes.

a. Show that (X1/n1)  (X2/n2) is an unbiased estimator for p1p2. [Hint: E(Xi)nipi for i1, 2.]

b. What is the standard error of the estimator in part (a)?

c. How would you use the observed values x1 and x2 to estimate the standard error of your estimator?

d. If n1  n2  200, x1  127, and x2  176, use the estimator of part

(a) to obtain an estimate of p1  p2.

e. Use the result of part

(c) and the data of part (d)
to estimate the standard error of the estimator.