“Have you ever stolen something worth more than $10?” Anyone asked this question in a survey might be reluctant to answer truthfully, especially if he or she did not wish to make known a past misbehavior. The respondent might be more willing to tell the truth if the questioner doesn’t know which of two questions, randomly chosen, the responder is answering (Warner 1965). This approach was used to estimate the true fraction of thieves among the third-year biology undergraduate population on a university campus in 2006. A total of 185 students participated. Each student was instructed to flip a coin and conceal the outcome. He or she was to respond with a yes if the outcome was heads. If the outcome was tails, the student was to answer the theft question truthfully with a yes or a no. The result: 113 of the 185 students responded with a yes, whereas the remaining 72 answered no.

Assume that students answered truthfully and independently that the probability of heads was 0.5, and that the sample of students was a random sample. Use likelihood to estimate the fraction of thieves in the student population.

a. Construct a probability tree (Chapter 5) to show that the probability of a student answering yes is (1+s)/2, where s is the fraction of thieves in the population.

b. If the assumptions are met, the number of yes answers in the survey of n students, Y, should follow a binomial distribution with the probability of a yes equal to (1+s)/2: Pr[Y _yeses | s] =(ⁿ_Y)(1+s/2) Y(1−s/2) ^n−Y. Write the formula for the likelihood of s, given the data.

c. Write the formula for the log-likelihood of s, given the data.

d. Calculate the log-likelihood that the fraction of thieves s is zero.