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 that 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.
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  = 0.25 + p/2 and hence use pÌ‚ = 2 - 0.5 to estimate p.

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Table for Randomized Response First Coin Head Tail Second Response Head 0.25 Tail 025 I-p2