Diamonds. Suppose that the couple shopping for a diamond in Example 28.15 (page 28-35) had used a quadratic regression model for the other quantitative variable, Depth. Use the data in Table 28.4 to answer the following questions.
(a) What is the estimated quadratic regression model for mean total price based on the explanatory variable Depth?
(b) As you discovered in part (a), it is always possible to fit quadratic models, but we must decide if they are helpful. Is this model as informative to the couple as the model in Example 28.15? What percent of variation in the total price is explained by using the quadratic regression model with Depth?
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.