28. In a study by Erich Brandt of 70 restaurant bills, 40 of the 70 were paid using cash. We assume a random sample and estimate the posterior distribution of the s/1n  15/118  3.5355 x

binomial parameter p, the population proportion paying cash.

a. Use a beta prior distribution with a  2 and b  2.

b. Use a beta prior distribution with a  1 and b  1.

c. Use a beta prior distribution with a and b very small and positive.

d. Calculate a 95% credibility interval for p using

(c). Is your interval compatible with p  .5?

e. Calculate a 95% con dence interval for p using Equation (8.10) of Section 8.2, and compare with the result of (d).

f. Calculate a 95% con dence interval for p using Equation (8.11) of Section 8.2, and compare with the results of

(d) and (e).

g. Compare the interpretations of the credibility interval and the con dence intervals.

h. Based on the prior in (c), test the hypothesis p

.5 using the posterior distribution to nd P(p

.5).