1. To get answers to sensitive questions, we sometimes use a method called the randomized
response technique. Suppose, for instance, that we want to determine what percentage of
the students at a large university take ketamine. We construct 20 flash cards, write ‘I take
ketamine at least once a week’ on 12 of the cards (where 12 is an arbitrary choice) and ‘I
do not take ketamine at least once a week’ on the others. Then we let each student (in the
sample interviews) select one of the 20 cards at random, and response yes or no without
divulging the question.
(a) Establish a relationship between P(Y ), the probability that a student will give a yes
response, and P(K), the probability that a student randomly selected at the university
takes ketamine at least once a week.
(b) If 110 of 250 students answered yes under these conditions, use the result of part (a)
and 110/250 as an estimate of P(Y ) to estimate P(K).
2. Four men and four women are ranked according to their scores on an aptitude test. Assume
that no two scores are alike and all 8! possible rankings are equally likely. Let X denote the
highest ranking achieved by a woman (for instance X = 2 if the top ranked person was male
and the next ranked person was female). Find P(X = i), i = 1, 2, · · · , 8. Hence find E(X)
and V ar(X).
response technique. Suppose, for instance, that we want to determine what percentage of
the students at a large university take ketamine. We construct 20 flash cards, write ‘I take
ketamine at least once a week’ on 12 of the cards (where 12 is an arbitrary choice) and ‘I
do not take ketamine at least once a week’ on the others. Then we let each student (in the
sample interviews) select one of the 20 cards at random, and response yes or no without
divulging the question.
(a) Establish a relationship between P(Y ), the probability that a student will give a yes
response, and P(K), the probability that a student randomly selected at the university
takes ketamine at least once a week.
(b) If 110 of 250 students answered yes under these conditions, use the result of part (a)
and 110/250 as an estimate of P(Y ) to estimate P(K).
2. Four men and four women are ranked according to their scores on an aptitude test. Assume
that no two scores are alike and all 8! possible rankings are equally likely. Let X denote the
highest ranking achieved by a woman (for instance X = 2 if the top ranked person was male
and the next ranked person was female). Find P(X = i), i = 1, 2, · · · , 8. Hence find E(X)
and V ar(X).