He Office of the Registrar at UCSD took a random sample of 427 students and obtained their grade point average in college (COLGPA), high school GPA (HSGPA), verbal Scholastic Aptitude Test scores (VSAT), and the mathematics scores in the SAT (MSAT). The following model was estimated (subscript t is omitted for simplicity):

COLGPA = B1 + B2 HSGPA + B3 VSAT + B4 MSAT + u

The estimated coefficients and their standard errors are given below:

           Coefficient     Standard error

B1            ignore      ignore

B2            0.398        0.061

B3          0.0007375    0.00028

B4          0.001015     0.0003 1.

1. The unadjusted R2 was 0.22. Because this is very low, we might suspect that the model is inadequate. Test the model for overall goodness of fit (using a 1 percent level of significance). Be sure to state the null and alternative hypotheses, the test statistic, its distribution, and the criterion for acceptance or rejection. What is your conclusion?

2. Test each regression coefficient for significance at the 1 percent level against the alternative that the coefficient is positive. Is any of them insignificant?

3. Suppose a student took a special course to improve her SAT scores and increased the verbal and math scores by 100 points each. On average, how much of an increase in college GPA could she expect from this?

4. Suppose you want to test the hypothesis that the regression coefficients for VSAT and MSAT are equal (but need not be equal to zero). Describe step-by-step how you should do this. State the null and alternative hypotheses, the regression(s) to be run, the test statistic to be computed, its distribution, and the criterion for accepting or rejecting the null hypothesis. What do you conclude?

5. List at least two other variables that should have been included in the model. Explain why you think they belong in the model.