In my textbook, a confidence interval of the mean response at some value is given by $$\hat{y}(x^0)\pm t_{\alpha/2}(n-p-1)s_p\sqrt{\frac{1}{n}+x_v^T(X^TX)^{-1}x_v} \ $$
, $n$ is number of data points, $p$ the number of coefficients
For a centered linear model and $p=1$ (i.e. two coefficients), however