This is a question that comes from An Introduction to Statistical Learning.

Suppose we have five predictors:

• $X_{1} = GPA$
• $X_{2} = IQ$
• $X_{3} = EDU$ (1 = College, 0 = High School)
• $X_{4} = GPA * IQ$
• $X_{5} = GPA * EDU$

Further, suppose that the response is starting salary, measured in thousands of dollars, and that we fit the following linear regression model:

$$ \begin{align} \hat{Y} & = \hat{\beta}_0 + \hat{\beta}_1 GPA + \hat{\beta}_2 IQ + \hat{\beta}_3 EDUC + \hat{\beta}_4 GPA + \hat{\beta}_5 GPA \times EDUC \\ \hat{Y} & = 50 + 20 \cdot GPA + 0.07 \cdot IQ + 35 \cdot EDUC + 0.01 \cdot GPA \times IQ - 10 \cdot GPA \cdot EDUC \end{align} $$

First, answer the following, True/False questions, and justify your answers:

1. For fixed IQ and GPA, HS grads earn more on average than college grads.
2. For fixed IQ and GPA, College grads earn more than HS grads.
3. For fixed IQ and GPA, HS grads earn more than college grads if GPA is high enough.
4. For fixed IQ and GPA, College grads earn more than HS grads if GPA is high enough.

Second, predict the salary of a college grad with IQ = 110 and GPA = 4.0.

Third, is the following statement True or False? Justify your answer.

Since the $GPA \times IQ$ coefficient is small, there is little evidence of interaction.