A comptroller was preparing to analyze the distribution of balances in the various ac counts receivable for her firm. She knew from studies in previous years that the distribu tion would be normal with a standard deviation of $1500, but she was unsure of the mean μ. She thought carefully about her uncertainty about this parameter and assessed a nor mal distribution for μ with mean m0 = $10,000 and σ0 = $800.
Over lunch, she discussed this problem with her friend, who also worked in the account- ing division. Her friend commented that she also was unsure of μ but would have placed it somewhat higher. The friend said that "better" estimates for m0 and σ0 would have been $12,000 and $750, respectively.
a. Find P(μ > $11,000) for both prior distributions.
b. That afternoon, the comptroller randomly chose nine accounts and calculated x = $11,003. Find her posterior distribution for /x. Find the posterior distribution of Μ for her friend. Calculate P(μ > $11,000) for each case.
c. A week later the analysis had been completed. Of a total of 144 accounts (including the nine reported in part b), the average was x- = $11,254. Find the posterior distribution for μ for each of the two prior distributions. Calculate P(Μ > $11,000) for each case.
d. Discuss your answers to parts a, b, and c. What can you conclude?