Suppose we are interested in estimating the fraction of students who have plagia

Question

Suppose we are interested in estimating the fraction of students who have plagiarized. Instead of asking an embarrassing question directly, we use the technique of "randomized response," as follows. We ask each student being surveyed to toss a coin in private. If the coin lands "heads," they are to answer the question, "Have you plagiarized a term paper?" truthfully. If the coin lands "tails," they are told always to answer "yes," whether they have in fact plagiarized or not. Imagine that the true fraction of students who have plagiarized is in fact 0.3, and imagine that participants in the survey indeed follow the randomized response procedure accurately and honestly. What is the conditional probability that a randomly chosen student who answers "yes" to the survey has in fact plagiarized?

Explanation / Answer

P(H) = P(T) = 0.5

Pr(Plagiarized) = 0.3

Pr(Someone say yes in the randomized survey response) = Pr(Plagrized who said yes) * Pr(Head) + Pr(plagrized who say yes) * Pr(Tail) = 0.3 * 0.5 + 0.5 = 0.65

so,

Pr(Palgrized l Yes to the survey) = 0.3 * 0.5/ 0.65 = 0.23

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