Compare the accuracies of the methods of Problems 28 and 29 by comparing their standard...

Compare the accuracies of the methods of Problems 28 and 29 by comparing their standard deviations. You may do this by substituting some plausible numerical values for p and q.

Reference

A variation of the method described in Problem 28 has been proposed. Instead of responding to statement 2, the respondent answers an unrelated question for which the probability of a “yes” response is known, for example, “Were you born in June?”

a. Propose an estimate of q for this method.

b. Show that the estimate is unbiased.

c. Obtain an expression for the variance of the estimate.

In surveys, it is difficult to obtain accurate answers to sensitive questions such as “Have you ever used heroin?” or “Have you ever cheated on an exam?” Warner (1965) introduced the method of randomized response to deal with such situations. A respondent spins an arrow on a wheel or draws a ball from an urn containing balls of two colors to determine which of two statements to respond to: (1) “I have characteristic A,” or (2) “I do not have characteristic A.” The interviewer does not know which statement is being responded to but merely records a yes or a no. The hope is that an interviewee is more likely to answer truthfully if he or she realizes that the interviewer does not know which statement is being responded to. Let R be the proportion of a sample answering Yes. Let p be the probability that statement 1 is responded to (p is known from the structure of the randomizing device), and let q be the proportion of the population that has characteristic A. Let r be the probability that a respondent answers Yes.

a. Show that r = (2p−1)q+(1−p). [Hint: P(yes)= P(yes given question 1)× P(question 1) + P(yes given question 2) × P(question 2).]

b. If r were known, how could q be determined?

c. Showthat E(R) = r , and propose an estimate, Q, for q. Showthat the estimate is unbiased.

d. Ignoring the finite population correction, show that Var(R) = r (1 − r )/ n where n is the sample size.

e. Find an expression for Var(Q).