Surveys and Probability Distribution

This is a very difficult question that I have been working on for 2 hours now.. Can any one help?

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. Suppose that 30% of the members of a particular population had incomes over $100,000 last year. A random sample of *n* = 10 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 he or she will give a truthful answer to this question is 0.7. If a person’s income was *not* over $100,000, the probability that he or she will give a truthful answer is 0.9.

*(a)* Find the probability distribution of *X*, the number of people in the sample who had incomes over $100,000 last year.

*(b)* Find the probability distribution of *Y*, the number of “yes” answers to the question in the sample of size 10.

*(c)* Find the probability distribution of *T*, the number of truthful answers from the 10 respondents.

*(d)* Given the answer “Yes, my income was over $100,000,” what is the probability that the income of this person was, indeed, over $100,000? Similarly, given the answer “No, my income was not over $100,000,” what is the probability that the income of this person was, indeed, below $100,000?

Re: Surveys and Probability Distribution

Re: Surveys and Probability Distribution

P(Income>100000) = 0.30

P(Truth|Income>100000) = 0.70

P(Truth|Income < 100000) = 0.90

a) X is Binomial(n=10, p=0.30)

b) Person says "yes" if

i)telling truth when: Income>100000 or Income<100000

ii)lying:Income>100000 or Income<100000

P("yes") = P(yes|Income>100000).P(Income>100000) + P(yes|Income<100000).P(Income<100000)

= P(Truth|Income>100000)*P(Income>100000) + P(Lied|Income<100000)*P(Income<100000)

= 0.70*0.30+0.10*0.70

=0.28

Y is Binomial (n=10,p=0.28)

c)

P(truth) = P(Truth|Income>100000)*P(Income>100000) +

P(Truth|Income<100000)*P(Income<100000)

= 0.70*0.30 + 0.90*0.70

=0.84

T is Binomial (n=10,p=0.84)

d) Using Bayes Theorem

P(Income>100000|"yes")

= P("yes"|Income>100000).P(Income>100000)/P("yes")

=0.7*0.30/0.28

=0.75