56. To encourage subjects to make responses on sensitive questions, the method of random- ized 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 ves and reporting the response tail if the truc response is no. Let z denote the true probability of the yes response on the sensitive question.

a) Explain why the numbers in Table 56 are the probabilities of the four possible out- comes

b) Let denote the sample proportion of subjects who report head for the second re- sponse Explain why =2p-5 estimates.

c) Using this approach, 200 subjects are asked whether they have ever knowingly cheated on their income tax. Report the estimate of r if the number of reported heads equals (i) 50, (ii) 70. (iii) 100. (iv) 150.

TABLE 5.6 Second Response First Coin Head Tail Head .25 .25 Tail 7/2 (1-x)/2