Question: A researcher wants to estimate the percentage of people in some population who have used illegal drugs, by conducting a survey. Concerned that a lot of people would lie when asked a sensitive question like "Have you ever used illegal drugs?", the researcher uses a method known as randomized response. A hat is filled with slips of paper, each of which says either "I have used illegal drugs" or "I have not used illegal drugs". Let p be the proportion of slips of paper that say "I have used illegal drugs" (p is chosen by the researcher in advance). Each participant chooses a random slip of paper from the hat and answers (truthfully) "yes" or "no" to whether the statement on that slip is true. The slip is then returned to the hat. The researcher does not know which type of slip the participant had. Let y be the probability that a participant will say "yes", and d be the probability that a participant has used illegal drugs.
(a) Find y, in terms of d and p.
(b) What would be the worst possible choice of p that the researcher could make in designing the survey? Explain.
(c) Now consider the following alternative system. Suppose that proportion p of the slips of paper say "I have used illegal drugs", but that now the remaining 1 p say "I was born in winter" rather than "I have not used illegal drugs". Assume that 1/4 of people are born in winter, and that a person's season of birth is independent of whether they have used illegal drugs. Find d, in terms of y and p.