Question: In the situation of problem, suppose that a second random sample of cardholders was selected, and 7 out of the 17 people in the sample were found to pay their bills on time. Construct the new posterior distribution containing information from the prior distribution and both samplings. Again, give a credible set of highest posterior probability close to 95%, and plot the posterior distribution.
Problem: Bank of America recently launched a special credit card designed to reward people who pay their bills on time by allowing them to pay a lower-than-usual interest rate. Some research went into designing the new program. The director of the bank's regular credit card systems was consulted, and she gave her prior probability distribution of the proportion of cardholders who would qualify for the new program. Then a random sample of 20 cardholders was selected and tracked over several months. It was found that 6 of them paid all their credit card bills on time. Using this information and the information in the following table-the director's prior distribution of the proportion of all cardholders who pay their bills on time-construct the posterior probability distribution for this parameter. Also give a credible set of highest posterior probability close to 95% for the parameter in question. Plot both the prior and the posterior distributions of the parameter.
Proportion Probability
0.1 0.2
0.2 0.3
0.3 0.1
0.4 0.1
0.5 0.1
0.6 0.1
0.7 0.1