To encourage subjects to make responses on sensitive questions, the method of randomized response is often used. The subject is asked to flip a coin, in secret.

If it is a head, the subject tosses the coin once more and reports the outcome, head or tails. If, instead, the first flip is a tail, the subject reports instead the response to the sensitive question, for instance, reporting the response head if the true response is yes and reporting the response tail if the true response is no. Let π denote the true probability of the yes response on the sensitive question.

(a) Explain why the numbers in Table 5.4 are the probabilities of the four possible outcomes.

(b) Let p denote the sample proportion of subjects who report head for the second response. Explain why πˆ =

2p − 0.5 estimates π.

(c) Using this approach, 200 subjects are asked whether they have ever knowingly cheated on their income tax.

Report the estimate of π if the number of reported heads equals (i) 50, (ii) 70, (iii) 100, (iv) 150.

TABLE 5.4 Second Response First Coin Head Tail Head 0.25 0.25 Tail π/2 (1− π)/2