The value of a diamond is determined by the four C's: carat weight, color, clarity, and cut. Carat weight is the standard measure for the size of a diamond. Generally, the more a diamond weighs, the more valuable it will be. The Gemological Institute of America (GIA) determines the color of diamonds using a 22-grade scale from D (almost clear white) to Z (light yellow). Colorless diamonds are generally considered the most desirable. The clarity of a diamond refers to how "free" the diamond is of imperfections and is determined using an 11-grade scale: flawless (FL), internally flawless (IF), very, very slightly imperfect (VVS1, VVS2), very slightly imperfect (VS1, VS2), slightly imperfect (SI1, SI2), and imperfect (I1, I2, I3). The cut of a diamond refers to the diamond's proportions and finish. Put simply, the better the diamond's cut is, the better it reflects and refracts light, which makes it more beautiful and thus more valuable. The cut of a diamond is rated using a five-grade scale: Excellent, Very Good, Good, Fair, and Poor. Finally, the shape of a diamond (which is not one of the four C's) refers to its basic form: round, oval, pear-shaped, marquis, and so on. A novice might confuse shape with cut, so be careful not to confuse the two. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 14_6_8 using the file format of your choice for the version of the text you are using.
The data represent a random sample of 40 unmounted, round-shaped diamonds. Use the data to answer the questions that follow:
(a) Determine the level of measurement for each variable.
(i) Carat weight
(ii) Color
(iii) Clarity
(iv) Cut
(v) Price
(vi) Shape
(b) Construct a correlation matrix. To do so, first convert the variables color, clarity, and cut to numeric values as follows:
Color: D = 1, E = 2, F = 3, G = 4, H = 5, I = 6, J = 7
Clarity: FL = 1, IF = 2, VVS1 = 3, VVS2 = 4, VS1 = 5,
VS2 = 6, SI1 = 7, SI2 = 8
Cut: Excellent = 1, Very Good = 2, Good = 3
If price is to be the response variable in our model, is there reason to be concerned about multicollinearity? Explain.
(c) Find the "best" model for predicting the price of a diamond.
(d) Draw residual plots, a boxplot of the residuals, and a normal probability plot of the residuals to assess the adequacy of the "best" model.
(e) For the "best" model, interpret each regression coefficient.
(f) Determine and interpret R2 and the adjusted R2.
(g) Predict the mean price of a round-shaped diamond with the following characteristics: 0.85 carat, E, VVS1, Excellent.
(h) Construct a 95% confidence interval for the mean price found in part (g).
(i) Predict the price of an individual round-shaped diamond with the following characteristics: 0.85 carat, E, VVS1 Excellent.
(j) Construct a 95% prediction interval for the price found in part (i).
(k) Explain why the predictions in parts (g) and (i) are the same, yet the intervals in parts (h) and (j) are different.