It is difficult to conduct sample surveys an sensitive issues because many peopl

Question

It is difficult to conduct sample surveys an sensitive issues because many people will not answer questions if the answers might embarrass them. Randomized response is an effective way to guarantee anonymity while collecting information on topics such as student cheating or sexual behavior. Here is the idea. To ask a sample of students whether they have plagiarized a term paper while in college, have each student toss a coin in private. If the coin lands heads and they have not plagiarized, they are to answer "No." Otherwise they are to give "Yes" as their answer. Only the student knows whether the answer reflects the truth or just the coin toss, but the researchers can use a proper random sample with follow-up for nonresponse and other good sampling practices. Suppose that in fact the probability is 0.3 that a randomly chosen student has plagiarized a paper. Draw a tree diagram in which the first stage is tossing the coin and the second is the truth about plagiarism. The outcome at the end of each branch is the answer given to the randomized-response question. What is the probability of a "No" answer in the randomized-response poll? If the probability of plagiarism were 0.26, what would be the probability of a "No" response on the poll? Now suppose that you get 36% "No" answer in a randomized-response poll of a large sample of students at your college. What do you estimate to be the percent of the population who have plagiarized a paper? % Slot machines are now video games, with winning determined by electronic random number generators. In the old days, slot machines were like this: you pull the lever to spin three wheels: each wheel has 20 symbols, all equally likely to show when the wheel stops spinning: the three wheels are independent of each other. Suppose that the middle wheel has 7 bells among its 20 symbols, and the left and right wheels have 1 bell each. (a) You win the jackpot if all three wheels show bells. What is the probability of winning the jackpot? (b) What is the probability that the wheels stop with exactly 2 bells showing?

Explanation / Answer

1) a) P(No) = P(Heads)*P(No) = 0.5*0.7 = 0.35

b) P(No) = 0.5*0.74 = 0.37

c) P(Heads)*P(Yes) = 0.14, P(Yes) = 0.14/0.5 = 0.37

2) b) The 2 bells could be 1st and 2nd wheel or 2nd and 3rd wheel or 1st and 3rd wheel.

P(exactly 2 bells) = (1/20 * 7/20) + (7/20 * 1/20) + (1/20 * 1/20) = 0.0375

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