In Chapter 19 we looked at data on the prices of diamonds (For Example: "Diamond prices"). Although higher carat weight of diamonds generally means higher cost, Carat Weight isn't the only factor that determines a diamond's price. A typical diamond is pale yellow. The less colour a diamond has, the higher its colour grade and-generally—its price. We want to build a model that incorporates both Carat Weight and Colour to predict a diamond's price. The data here are a collection of 749 diamonds: Weight is between 0.3 and 1.4 carats; colour grades are D (highest possible), G (medium), and K (fairly low). We'll use Log \({ }_{10}\) Price as the response to make the relationship more linear. Here are scatterplots of Price and Log \({ }_{10}\) Price vs. Carat Weight for these diamonds:

A linear model for \(\log _{10}\) Price on Carat Weight alone finds:
Response Variable: \(\log _{10}\) Price
\(R^{2}=77.39 \%\) Adjusted \(R^{2}=77.36 \%\)
\(s=0.1364\) with \(749-2=747\) degrees of freedom

Here is a scatterplot of \(\log _{10}\) Price vs. Carat Weight showing the three different colour grades (D, G, and K):

To account for the differences, two indicator variables were created:
\[\begin{aligned}
& \text { Colour } D=1 \text { if Colour }=\mathrm{D} \text { and } 0 \text { otherwise } \\
& \text { Colour } G=1 \text { if Colour }=\mathrm{G} \text { and } 0 \text { otherwise }
\end{aligned}\]
A multiple regression of \(\log _{10}\) Price was run with three predictors: Carat Weight, Colour D, and Colour G.

Here is the output:
Response Variable: \(\log _{10}\) Price
\(R^{2}=85.66 \%\) Adjusted \(R^{2}=85.61\)
\(s=0.1088\) with \(749-4=745\) degrees of freedom

QUESTIONS:

 For the three colours of diamonds, what are the models that predict \(\log _{10}\) Price? Does the addition of the indicator variables for the colours seem like a good idea? Why are there only two indicator variables?

Transcribed Image Text:

10,000 4.0 9,000 8,000 3.8 7,000 3.6 6,000 5,000. 3.4 4,000 3,000 3.2 2,000 1,000 3.0 ++ 0.4 0.6 0.8 1 1.2 1.4. 0.4 0.6 0.8 1 Carat Weight Carat Weight ++ 1.2 1.4