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2. [-/4 Points] DETAILS DEVORESTAT9 6.E.019. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 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 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 ) = 0.5p + 0.15 for p and then substitute Y for ).] (Enter your answer in terms of Y and n.) p = If n = 80 and y = 28, what is your estimate? (b) Use the fact that E(Y) = 1 to show that your estimator p is unbiased. This answer has not been graded yet. (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 =1. [-/12 Points] DETAILS DEVORESTAT9 5.E.001.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. p(x, y) 0 1 2 0 0.10 0.05 0.01 1 0.07 0.20 0.07 2 0.05 0.14 0.31 (a) What is P(X = 1 and Y = 1)? P(X = 1 and Y = 1) = (b) Compute P(X S 1 and Y S 1). P(X S 1 and Y S 1) = (c) Give a word description of the event {X # 0 and Y # 0}. One hose is in use on one island. At least one hose is in use at both islands. At most one hose is in use at both islands. One hose is in use on both islands Compute the probability of this event. P(X # 0 and Y # 0) = (d) Compute the marginal pmf of X. 0 2 x(x) Compute the marginal pmf of Y. 1 2 pyly Using py(x), what is P(X S 1)? P(X S 1) = (e) Are X and Y independent rv's? Explain. O X and Y are not independent because P(x,y) = Px(x) . Py(y). O X and Y are independent because P(x,y) = Px(x) . Py(y). O X and Y are independent because P(x,y) # Px(x) . Py(y). X and Y are not independent because P(x,y) # Px(x) . Py(y).B. [1/2 Points] PREVIOUS ANSWERS DEVDRESTAT9 5.E.053.S. Rockwell hardness of pins ofa certain type is known to have a mean value of 50 and a standard deviation of 1.5. (Round your answers to four decimal places.) l use SALT (a) If the distribution is normal. what is the probability that the sample mean hardness for a random sample of 18 pins is at least 51? 0.2236 x (b) What is the (approximate) probability that the sample mean hardness for a random sample of 4D plns Is at least 51.7 0.0001 J
