Diamonds. Consider the diamond data of which Table 28.4 (page 28-35) is an excerpt. We are interested in predicting the total price of a diamond. Fit a simple linear regression model using Carat as the explanatory variable.
(a) Identify the least-squares line for predicting Total Price from Carat.
(b) Does the model provide a good fit? Comment on the residual plots. How much variation in price can be explained with this regression line?
(c) Create a new variable Caratsq = Carat × Carat. Fit a quadratic model using Carat and Caratsq and verify that your estimates for each parameter match those provided in Example 28.15 (page 28-35).
(d) Does the quadratic term Caratsq improve the fit of the model? Comment on the residual plots and the value of R2 .
(e) The individual t statistics look at the contribution of each variable when the other variables are in the model. State and test the hypotheses of interest for the quadratic term in your model.
Example 28.15:
If there is a quadratic relationship between a quantitative variable y and another quantitative variable x1, the mean response is given by
A young couple are shopping for a diamond, so they are interested in learning more about how these gems are priced. They have heard about the 4 C's: carat, color, cut, and clarity. Is there is a relationship between these diamond characteristics and the price? Table 28.4 shows records for the first 10 diamonds in a large data base.6 The complete data base contains 351 diamonds and is available in the file DIAMONDS.dat. The variables include Carat, Color, Clarity, the Depth of the cut, the price per carat Price/Ct, and the Total Price. Since the young couple are primarily interested in the price of a diamond, they decide to begin by examining the relationship between Total Price and Carat. Figure 28.11 shows a scatterplot of Total Price versus Carat, along with the estimated quadratic
FIGURE 28.11:
regression model. Using the quadratic regression model, the couple estimate the mean price of a diamond to be
The couple are happy because they can explain 92.6% of the variation in the total price of the diamonds in the data base using this quadratic regression model. However, they are concerned because they used explanatory variables that are not independent. An explanatory variable and its square are obviously related to one another. The correlation between Carat (x1) and Carat2 (x12) is 0.952.