The file P13_66.XLS 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 ignored the two explanatory variables MachHrs and DirMatCost and predicted each OHCost as the mean of all OHCosts, then a typical “error” would be OHCost minus the mean of all OHCosts. Find the sum of squared errors using this form of prediction, where the sum is over all observations. 

c. Now run three regressions: (1) OHCost versus MachHrs, (2) OHCost versus DirMatCost, and (3) OHCost versus both MachHrs and DirMatCost. (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 response 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, not just the correlations between the response 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 OHCost. Verify that this is the case for these data.