dolution

The file P10_66.xlsx contains monthly cost accounting data on overhead costs, machine hours, and direct material costs. This problem will help you explore the meaning of R₂ and the relationship between R₂ and correlations.

a. Create a table of correlations between the individual variables.

b. If you ignore the two explanatory variables Machine Hours and Direct Material Cost and predict each Overhead Cost as the mean of Overhead Cost, then a typical “error” is Overhead Cost minus the mean of Overhead Cost. Find the sum of squared errors using this form of prediction, where the sum is over all observations.

c. Now run three regressions: (1) Overhead Cost (OH Cost) versus Machine Hours, (2) OH Cost versus Direct Material Cost, and (3) OH Cost versus both Machine Hours and Direct Material Cost. (The first two are simple regressions, the third is a multiple regression.) For each, find the sum of squared residuals, and divide this by the sum of squared errors from part b. What is the relationship between this ratio and the associated R₂ for that equation? (Now do you see why R₂ is referred to as the percentage of variation explained?)

d. For the first two regressions in part c, what is the relationship between R₂ and the corresponding correlation between the dependent and explanatory variable? For the third regression it turns out that the R₂ can be expressed as a complicated function of all three correlations in part a. That is, the function involves not just the correlations between the dependent variable and each explanatory variable, but also the correlation between the explanatory variables. Note that this R₂ is not just the sum of the R₂ values from the first two regressions in part c. Why do you think this is true, intuitively? However, R2 for the multiple regression is still the square of a correlation—namely, the correlation between the observed and predicted values of OH Cost. Verify that this is the case for these data.

Q689

The file P10_67.xlsx contains hypothetical starting salaries for MBA students directly after graduation. The file also lists their years of experience prior to the MBA program and their class rank in the MBA program (on a 0−100 scale).

a. Estimate the regression equation with Salary as the dependent variable and Experience and Class Rank as the explanatory variables. What does this equation imply? What does the standard error of estimate se tell you? What about R₂?

b. Repeat part a, but now include the interaction term Experience*Class Rank (the product) in the equation as well as Experience and Class Rank individually. Answer the same questions as in part a. What evidence is there that this extra variable (the interaction variable) is worth including? How do you interpret this regression equation? Why might you expect the interaction to be present in real data of this type?