Question 7.51
21. Randomized response: Suppose 50 students in a college class participate in a survey in which they each flip a coin and do not reveal the result. If the result is tails, the student is supposed to give an honest answer to the question “Have you ever cheated on an exam in high school or in college?” If the result is heads, the student is supposed to say “yes” to that question, regardless of what the true answer is. Suppose the results in the class are 42 “yes” answers and 8 “no” answers.
- If students follow the procedure correctly, is it true that all students who answered “no” have not cheated on an exam either in high school or in college?
- If students follow the procedure correctly, is it true that all students who have not cheated on an exam either in high school or in college answered “no”?
- On average, about half of the students who have not cheated on an exam in high school or in college flipped tails, so what is your best estimate of the true number of students who have not cheated on an exam?
- Based on the answer to part (c), what is your estimate of the true number and proportion of students who have cheated on an exam either in high school or in college?
- Do we have any way to know which of the 42 “yes” answers are truthful?
21.
Sample response: (a) Yes. The only way a student can answer “no” is if the coin landed on tails, after which the student would have to answer the question honestly.
(b) Even though a student has not cheated, if the result of the coin flip is heads, then the student must answer “yes.” So the statement” It is true that all students who have not cheated on an exam in high school or in college answered ‘no’” would be a false statement.
(c) About 16 students have not cheated on an exam in high school or in college.
(d)
(e) No. There is no way of telling which of the “yes” answers are true.