Ofn₁randomly selected male smokers,X₁smoked filter cigarettes, whereas ofn₂randomly selected female smokers,X₂smoked filter cigarettes. Letp₁andp₂denote the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes.

(a) Show that(X₁/n₁)(X₂/n₂) is an unbiased estimator forp₁p₂. [Hint:E(X_i) =n_ip_ifori= 1, 2.]

EX₁

n₁

X₂

n₂

=1

n₁

E1

n₂

E

=1

n₁

1

n₂

=p₁p₂

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

(c) How would you use the observed valuesx₁andx₂to estimate the standard error of your estimator?

You would use them to estimate the necessary quantities and then put these into the formula.

You would put them directly into the formula to obtain the estimate of the standard error.

You can not use the observed values to estimate the standard error of your estimator.

There is not enough information.

(d) Ifn₁=n₂=208,x₁=131, andx₂=162, use the estimator of part (a) to obtain an estimate ofp₁p₂. (Round your answer to three decimal places.)

(e) Use the result of part (c) and the data of part (d) to estimate the standard error of the estimator. (Round your answer to three decimal places.)