a. Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?

b. Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Ryan wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?

c. Suppose Ryan decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

d. Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Ryan’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Ryan purchase?

e. Ryan now remembers that it is sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Ryan wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?

f. Suppose Ryan decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

g. Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Ryan’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actually price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Ryan purchase?

h. Ryan now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Ryan wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?

i. Suppose Ryan decides to use clarity (X1), carats (X3) and the interaction terms X4 and X5 as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

j. Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Ryan’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Ryan purchase?

With Christmas coming, Ryan Bellison was searching for the perfect gift for his wife. Ryan leaned back in his chair at the office and tried to think, after several years of marriage, of the one thing his wife had wanted during the years they pinched pennies to get through graduate school. Then he remembered the way her eyes had lit up last week when they walked by the jewelry store windows at the mall and she saw the diamond earrings. He knew he wanted to see that same look on her face Christmas morning. And so his hunt began for the perfect set of diamond earrings.

Ryan’s first order of business was to educate himself about the things to look for when buying diamonds. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.

After a bit of searching, Ryan located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably and he wanted to make sure he didn’t get ripped off. To assist in his decision making, Ryan decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls on your data disk for this purpose. Use this data to answer the following questions for Ryan.