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

a. Show that (X1/n1) (X2/n2) is an unbiased estimator for p1 p2. [Hint: E(Xi) ni pi for i 1, 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.