6.6 Montgomery, et al. (2000, p. 75) give n = 25 observations on the amount of time taken to service vending rnacliines by the route driver. The amount of time y (measured in minutes) is associated with the number of cases of product stocked (xl) and the distance in feet walked by the route driver (x2). The full data is given in Table 6.5.

(a) Suppose that we ignore the covaniates for the time being and model y with the model given in Problem 6.5. Fix the hyperparameters ofthc nomal and inverse-gamma prior distributions.

(b) Use your results from Problem 6.5 to sample the posterior density of

@,

d) given the data on y = ( y , .y2, . . . ,.vzs).

(c) Now consider the Bayesian multiple regression model Gibbs sampler to sample the posterior density h@. a2 1) mskzply).

where xi = ( 1 , xl,.x 2,). You can fix the hyperpardmeter values (B0,B ,) and (vO,6 ,)

as you see fit. Use the full conditional densities given in the chapter and the Gibbs sampler to obtain M = 5000 drdws fiom the posterior density h@, 2 I y). Use the sampled draws to estimate the posterior mean and standard deviations of /? and 02.

(d) Given your sampling scheme in part (c), use the method of composition to obtain M = SO00 draws from the predictive density under the assumption that xi6 = (1,6.500). Use the sampled draws to estimate the predictive mean and standard deviation.