4. In a survey, some of the questions concern sensitive issues (e.g., income, drug use, sexual experiences). As a result, some respondents do not answer the questions truthfully. Denote the proportion of the members of a particular population that had incomes over $100,000 last year by p. A random sample of n members of this population is taken, and each person in the sample is asked “Was your income over $100,000 last year?” If a person really had an income over $100,000, the probability that she will give a truthful answer to this question is 1-1. If a person’s income was not over $100,000, the probability that she will give a truthful answer is 1-2. From past experience, 1 and 2 are known, with 0<1<0.5, 0<2<0.5.

a) For a sample of size one, find the likelihood function if the answer is “yes” and find the likelihood function if the answer is “no.”

b) For a random sample of size n, find the likelihood function and sufficient statistics.

c) Find the maximum likelihood estimator for p.

d) Assume that 1=0.1, 2=0, and there is one “yes” answer in a random sample of size 10. What is your best estimate of p and why?

e) Consider the same scenario as in (d), but assume that 1 is unknown (0<1<1). In this case, what would be your best estimate of p and why?

6. A journal editor says: “If we only publish papers with results that are statistically significant at the =0.05 level, at most 5% of our papers will have erroneous results.” Denote by p the proportion of researchers with true H0 and false H1. Suppose that each researcher performs one test, sends the paper to the journal, and the paper is accepted if the results of the test are significant at the =0.05 level.

a) If in a given year the journal publishes n papers, find the distribution of the papers with erroneous results that are published in this year. Assume that all the tests in all papers have the same , probability of type II error.

b) What is this distribution if p=1, i.e., if all researchers, submitting the papers this year, had true H0 and false H1?

c) Overall, comment on the above statement of a journal editor.