11.20 Testing a collection of variables. Refer to the previous exercise. For the model that included all p = 4 explanatory variables, only InCostAid is found significant using the individual parameter t tests. This raises the question whether these other three variables further contribute to the prediction of average debt given in-state cost is in the model.

In this chapter, we discussed the F test for a collection of regression coefficients. In most cases, this capability is provided by the software. When it is not, the test can be performed using the R²-values from the larger (full) and smaller (reduced) models. The test statistic is

with q and n − p − 1 degrees of freedom. is the value for the full model, and is the value for the reduced model. Here n = 24 schools, p = 4 variables in the full model, and q = 3 variables were removed to form the reduced model. Plug in the values of R² from part (b) of the previous exercise and the R² value from Figure 10.13 (page 580). Compute the test statistic and P-value. Do Admit, GradRate, and OutCostAid combined add any significant predictive information beyond what is already contained in InCostAid?