Answer: An investigator wishes to estimate the proportion

Chapter 6, Problem 6.19

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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 P(yes response). Then and p are related by .5p (.5)(.3). a. Let Y denote the number of yes responses, so Y Bin (n, ). Thus Y/n is an unbiased estimator of . Derive an estimator for p based on Y. If n 80 and y 20, what is your estimate? [Hint: Solve .5p .15 for p and then substitute Y/n for .] b. Use the fact that E(Y/n) to show that your estimator p is unbiased. c. If there were 70 type I and 30 type II cards, what would be your estimator for p?