8.128 Randomized response To encourage subjects to make honest responses on sensitive questions, the method of randomized response is often used. Let’s use your class to estimate the proportion who have had alcohol at a party. Before carrying out this method, the class should discuss what they would guess for the value of the proportion of students in the class who have had alcohol at a party.

Come to a class consensus. Now each student should flip a coin in secret. If it is a head, toss the coin once more and report the outcome, head or tails. If the first flip is a tail, report instead the response to whether you have had alcohol, reporting the response head if the true response is yes and reporting the response tail if the true response is no. Let p denote the true probability of the yes response on the sensitive question.

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a. Explain why the numbers in the following table are the probabilities of the four possible outcomes. b. Let denote the sample proportion of subjects who report head for the second response. Explain why we can set q = 0.25 + p/2 and hence use p = 2q - 0.5 to estimate p. c. Using this approach with your class, estimate the probability of having had alcohol at a party. Is it close to the class guess? Table for Randomized Response First Coin Head Tail Second Response Head Tall 0.25 p/2 0.25 (1 - p)/2