Refer to Practice Problem 5.

a. Using a spreadsheet or other program, calculate the log-likelihood of values of s between 0.0 and 0.5 in increments of 0.01. Using this information, determine the maximum likelihood estimate of the fraction of thieves.

b. Using the same approach as in part (a), calculate the likelihood-based 95% confidence interval for the parameter s.

c. Are there truly thieves among us? Using the values for the likelihood calculated in your answers to Practice Problem 5, use the log-likelihood ratio test to test the null hypothesis that s is zero.

Data from Practice Problem 5

“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.