Can I apply the t-test to 2 variable linear regression?

Question

Suppose I can predict the GPA of a college freshman's first semester using their high school grades with a simple linear regression, y = ax + b, where y is either the actual or predicted first semester college GPA, x is there high school gpa and b is an intercept. Suppose I have this data, can find there are optimal values for a and b, and high school grades are indeed predictive of college grades.

Suppose my hypothesis is that grades in math are more predictive of success than grades in other subjects. If I have all the data, I can fit another linear regression y = ax1 + bx2 + c, where x1 is high school math grades and x2 is high school grades of all other subjects. Suppose I do this, and optimal values fitting my regression show a higher coefficient for x1 (math) than x2 (other subjects).

This does not prove my hypothesis since I started with the assumption high school grades were already predictive but not a perfect correlation, e.g. I already know some high schools students with straight A's do worse than some B students. What I want to do is fit a p-value to my hypothesis showing the probability that math is actually more predictive than other subjects for my data set, and I didn't just get a higher coefficient based on noise.

How do I do this? My initial assumption was to use Student's t-test, but I don't have 2 populations (every single student took math and other subjects before college). I could partition my data set, top half of students in math vs the rest, but this seems like the wrong way to go about the problem, and I'd have to deal with the confounding variable of how students did in other subjects, since that is definitely not an independent variable.

Answer 1

Rather than a t-test, you can compare two models using ANOVA as described in this post.

You could test any give subject's grades vs. the base model of overall grades to see if they improve the residual sum of squares. Or, starting with the model with overall and math grades, add other subject's grades and see if the model improves. This will give you and indication of whether other subjects are able to capture some variance that overall grades and math grades did not, but it doesn't necessarily tell you the importance.