c. (5 Points) Plot the residuals versus x. Do you detect any trends? If so, what does the pattern suggest about the model? Copy and paste the plot of the residuals versus x below. Notice that the plot reveals a quadratic U-shaped trend. For small and large values of X, the residuals are (positive or negative), while for intermediate values of x , the residuals are (positive or negative). This pattern indicates a lack-of-fit in the straight-line model and suggests that a term to introduce in the model is necessary. To verify this, we would fit the quadratic model Ely =8,+8,x+8,x and test for curvature, i.e., test Ho: B2 =1 2. (10 Points) 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 ) ,*Bixte where y is the demand (in thousands) and * is the retail price per carat (dollars). a. (8 Points) Fit this model to the 12 data points given in the file GEM, and find the regression residuals. The fitted model is y= The residuals are