I’m a little unclear about what exactly a T-value in a regression means, and how it differs from the P value. And what does a negative T-value mean-- particularly a large one?
It means the same thing as a positive one. If the smaller mean is subtracted from the larger one, it is positive, but it is negative if opposite. The sign tells you the directionality of the relationship. So you would get significant results with both 4.5 and -4.5, it just tells you which mean is larger.
The test statistic (t in this case) and p supplement each other. The t needs to be greater than the critical value, which tells you the difference is extreme enough to be significant. The p-value tells you the probability of getting a more extreme result, given that the null is true. To oversimplify, a p =.03 would tell you that there is about a 3% chance that your results were obtained purely by chance. A significant p needs to be less than alpha (usually .05).
As t goes up, p goes down, and vice versa.