In the 1960s, Warner and Greenberg proposed methods for maintaining respondent confidentiality on a survey question by using randomized responses. randomized responses. When a question concerns a sensitive subject, for example criminal behavior, the respondent is not required to answer. criminal behavior, the respondent may not be honest. If we can convince the respondent that the answer will be confidential, we hope the respondent will be honest. Warner suggested that a question about a sensitive topic be phrased as a Yes/No question and that the respondent also be offered the negation of the question. Suppose we want to estimate the prevalence of bullying in a high school. We could ask the student the following two questions:

(A) I have bullied a classmate in the past two years. (Yes/No) (B) I have not bullied a classmate in the past two years. (Yes/No)

We ask the respondent to toss a coin (or roll a dice) and depending on the result of the toss result of the toss, we ask that the respondent give the answer to (A) or (B). We will not know the result of the toss and so we will not know if the respondent is answering to version (A) or version (B) of the question. For the following questions, we assume that the respondent rolls a die and must answer (A) of the question only if the result of the roll is 1. If the result of the If the result of the roll is not 1, the respondent will answer version (B). Let us assume that the respondents are honest.

(a) Suppose that the probability that the respondent has bullied a classmate in the last two years is the past two years is 13%. What is the probability that the respondent will answer Yes to the question? (b) Suppose the probability that the respondent has bullied a classmate in the past two years is 13%. 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 of the 395 respondents answered Yes to the question. Give an approximate the probability that the respondent has bullied a classmate in the last two years. in the last two years.

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