Use the data on dishwasher shipments in Table 11.13 on page 744. Suppose that we wish to fit a multiple linear regression model for predicting dishwasher shipments from time (year minus 1960) and private residential investment. Suppose that the parameters have the improper prior proportional to 1/τ. Use the Gibbs sampling algorithm to obtain a sample of size 10,000 from the joint posterior distribution of the parameters.
a. Let β1 be the coefficient of time. Draw a plot of the sample c.d.f. of |β1| using your posterior sample.
b. We are interested in predicting dishwasher shipments for 1986.
i. Draw a histogram of the values of β0 + 26β1 + 67.2β2 from your posterior distribution.
ii. For each of your simulated parameters, simulate a dishwasher sales figure for 1986 (time = 26 and private residential investment = 67.2). Compute a 90 percent prediction interval from the simulated values and compare it to the interval found in Example 11.5.7.
iii. Draw a histogram of the simulated 1986 sales figures, and compare it to the histogram in part i. Can you explain why one sample seems to have larger variance than the other?