I'm learning about regression. In one exercise I was asked some questions about the following output in R giving information about a linear model concerning housing prices in Boston with lots of covariates.
I don't have previous experience with R, but I'm also a newbie to the theory behind linear regression. We have our model $y=X\beta + \epsilon$, and we've made our fit $\hat{y}=X\hat{\beta}$.
I'm fairly confident that the "Estimate" column refers to the corresponding coordinate of $\hat{\beta}$. The standard error I believe(?) refers to the sample deviation of the corresponding covariate, so $\sqrt{\frac{ \sum_j ((X_1)_j-\overline{X_1})^2 }{n-1}}$, in the case of the first covariate, for example, although I am not sure.
My question is, what student statistic is the t-value referring to, and why is it a student random variable? I've looked up many articles but they are all frustratingly hand-wavy about this type of R output. Even the similar Mathexchange posts I saw didn't help me out.