Demand for a rare gem. A certain type of rare gem serves as a status symbol for many of its owners. In theory, then, the demand for the gem would increase as the price increases, decreasing at low prices, leveling off at moderate prices, and increasing at high prices, because obtaining the gem at a high price confers high status on the owner. Although a quadratic model would seem to match the theory, the model proposed to explain the demand for the gem by its price is the first-order model y = β0 + β1x + ε
where y is the demand (in thousands) and x is the retail price per carat (dollars). This model was fit to the 12 data points given in the table, and the results of the analysis are shown in the SAS printout (p. 398).
(a) Use the least squares prediction equation to ver- ify the values of the regression residuals shown on the printout.
(b) Plot the residuals against retail price per carat, x.
(c) Can you detect any trends in the residual plot? What does this imply?