webassign.net + Module 10 - Graded Assignment - STAT. C Let X Denote The Proportion Of Allotted. C A Diagnostic Test For A Certain Disease I.. C An Investigator Wishes To Estimate The An investigator wishes to estimate the proportion of students at a certain university who have violated the nonor code. Having obtained a random sample o 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 1 = P(yes response). Then 1 and p are related by 1 = 0.5p + (0.5)(0.3). (a) Let Y denote the number of yes responses, so Y ~ Bin(n, 1). Thus Y is an unbiased estimator of 1. Derive an estimator for p based on Y. [Hint: Solve 1 = 0.5p + 0.15 for p and then substitute Y for 1.] (Enter your answer in terms of Y and n.) p = 2y - 3 n X If n = 80 and y = 24, what is your estimate? (b) Use the fact that E(Y) = 1 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? (Enter your answer in terms of Y and n.) P =