In previous years, students in this course collected data on peoples preferences in the two Allais gambles below.

Gamble 1:

A :$2500 with probability 0.33

$2400 with probability 0.66

$0 with probability 0.01

B :$2400 with certainty $2400

Gamble 2:

C :$2500 with probability 0.33 $0 with probability 0.67

D :$2400 with probability 0.34 $0 with probability 0.66

For this problem, we will assume that responses are independent and identically distributed, and the probability is p that a person chooses both B in the first gamble and C in the second gamble. Comment on your results.

a. (5 pts) Assume that the prior distribution for p is Beta(1, 3). Find the prior mean and standard deviation for p. Find a 95% symmetric tail area credible interval for the prior probability that a person would choose B and C.

b. (10 pts) In 2009, 19 out of 47 respondents chose B and C. Find the posterior distribution for the probability p that a person in this population would choose B and C. Find the posterior mean and standard deviation, and a 95% symmetric tail area credible interval for p. Do a triplot.

c. (20 pts) Do you think Beta(1, 3) is a reasonable prior distribution to use for this problem? Why or why not? Give a comparison of using a Jeffereys prior vs using Beta(1, 3). Argue with mathematically-sound arguments NOT simply descriptive statements. 1

d. (25 pts) Find the predictive distribution for the number of responses choosing B and C in a future sample of 50 people drawn from the same population. Compare with a Binomial distribution using a point estimate of the probability of choosing B and C.

e. (25 pts) Among the 47 respondents, there are 20 males of whom 12 chose B and C. Use the prior in part (a) and a Bayesian solution to assess whether the probability of choosing B and C is higher for males than that for females?

f. (15 pts) Does the data suggest that there is no difference in choosing B and C between males and females? Use a Bayesian posterior predictive check to answer this question.