Suppose we want to do [[hypothesis testing]] for $\begin{align} H_0 : \beta_j = c && H_1 : \beta_j \ne c \end{align} $ at significance level $\alpha$. Because the response $\vec Y$ is assumed to have a [[normal distribution]] and that $\sigma$ is unknown, the test statistic $t$ will have a [[t-distribution]] with $n-(p+1)$ [[degrees of freedom]]. The test statistic is $t(\vec Y) = \frac{\hat \beta_j - c}{\hat {s.e.} (\hat \beta_j)} \sim t\Big(n - (p+1) \Big )$ where $\hat {s.e.}(\beta_j) = \sqrt{\Big [\hat \sigma^2(X^TX)^{-1}]_{jj}}$ and where $\hat \sigma^2 = \frac{RSS}{n - (p+1)}$ In [[R]] the default null hypothesis is that the parameter is equal to $0$. Therefore, the `t value` reported by R is simply the estimate $\beta_j$ divided by the standard error. However, do not use the individual t-tests to decide which parameters to include in the model.