Exercise 8: Categorical Variables

Note

Suppose we have a data set with five predictors, \(X_1 = \text{GPA}\), \(X_2 = \text{IQ}\), \(X_3 = \text{Level}\) (1 for College and 0 for High School), \(X_4 =\) Interaction between GPA and IQ, and \(X_5 =\) Interaction between GPA and Level. The response is starting salary after graduation (in thousands of dollars). Suppose we use least squares to fit the model, and get \(b_0 = 50\), \(b_1 = 20\), \(b_2 = 0.07\), \(b_3 = 35\), \(b_4 = 0.01\), \(b_5 = −10.\).

1. Which answer is correct, and why?

   1. For a fixed value of IQ and GPA, high school graduates earn more, on average, than college graduates.

   2. For a fixed value of IQ and GPA, college graduates earn more, on average, than high school graduates.

   3. For a fixed value of IQ and GPA, high school graduates earn more, on average, than college graduates provided that the GPA is high enough.

   4. For a fixed value of IQ and GPA, college graduates earn more, on average, than high school graduates provided that the GPA is high enough.

2. Predict the salary of a college graduate with IQ of 110 and a GPA of 4.0.

3. True or false: Since the coefficient for the GPA/IQ interaction term is very small, there is very little evidence of an interaction effect. Justify your answer.