Some respondents give false responses to survey questions. Denote the proportion of the members of a particular population that smoked more than 50 cigarettes last week byp. A random sample ofnmembers of this population is taken, and each person in the sample is asked “Did you smoke more than 50 cigarettes last week?” If a person really smoked more than 50 cigarettes, the probability that he will give a truthful answer to this question is 1-x_{1}. If a person did not smoke more than 50 cigarettes, the probability that he will give a truthful answer is 1-x_{2}. From past data, x_{1}and x_{2}are known, with 0<x_{1}<0.5, 0<x_{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 sizen, find the likelihood function and sufficient statistics.

c) Find the maximum likelihood estimator forp.

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

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