An investigator wishes to estimate the proportion of students at a certain university who have violated the honor code. Having obtained a random sample of n students, she realizes that asking each, “Have you violated the honor code?” will probably result in some untruthful responses. Consider the following scheme, called a randomized response technique. The investigator makes up a deck of 100 cards, of which 50 are of type I and 50 are of type II.
Type I: Have you violated the honor code (yes or no)?
Type II: Is the last digit of your telephone number a 0, 1, or 2 (yes or no)?
Each student in the random sample is asked to mix the deck, draw a card, and answer the resulting question truthfully. Because of the irrelevant question on type II cards, a yes response no longer stigmatizes the respondent, so we assume that responses are truthful. Let p denote the proportion of honor-code violators (i.e., the probability of a randomly selected student being a violator), and let λ = P(yes response). Then l and p are related by λ = .5p + (.5)(.3).
a. Let Y denote the number of yes responses, so Y ∼ Bin (n, l). Thus Y/n is an unbiased estimator of λ. Derive an estimator for p based on Y. If n = 80 and y = 20, what is your estimate? [Hint: Solve λ = .5p + .15 for p and then substitute Y/n for λ.]
b. Use the fact that E(Y/n) = λ to show that your estimator is unbiased.
c. If there were 70 type I and 30 type II cards, what would be your estimator for p?
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