Problem 3.6 In the 1960s, Warner and Greenberg proposed ways to maintain the confidentiality of the respondent on a survey question by using a technique called randomized responses. If a question concerns a sensitive issue (e.g. criminal behavior), the respondent might not be truthful. If we can convince the respondent that the answer will be confidential, then this person may answer truthfully. Warner proposed that a question concerning a sensitive issue be formulated as a question with Yes/No question and that the respondent be offered the question and its negation. Suppose that we would like to estimate the prevalence of bullying in a high school. We then ask the student the following two questions:

(A) Have you bullied a classmate in the last two years? (Yes/No)

(B) Have you not bullied a classmate in the last two years? (Yes/No)

We ask the respondent to throw a die and answer version (A) of the question if the result on the die is 1, and version (B) otherwise. Since we do not know the result on the die, we cannot tell if the respondent has answered version (A) or version (B) of the question. Assume that the respondents are answering truthfully. (a) Suppose that the probability that a respondent has bullied a classmate in the last two years is 13%. What is the probability that the respondent will answer "Yes" to the question. (b) Suppose that the probability that a respondent has bullied a classmate in the last two years is p. Express p as a function of the probability of answering "Yes" to the question. (c) Suppose that 275 out of 395 respondents answered "Yes" to the question. Approximate the probability that a respondent has bullied a classmate in the last two years.