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

$$(X^\top X)^{-1}= \begin{pmatrix} \frac{1}{n} & 0 \\ 0 & \frac{1}{\sum_{i=1}^n (x_i-\bar{x})^2 }\end{pmatrix}$$

So the matrix under square root must be a truncated matrix, or?

This regards a question of mine on the main site, which someone has answered, see stats.stackexchange.com/a/507577/263915

@schn Please ask this question on the main site, which has better facilities for questions and answers.

@Sycorax I did ask it on the main site, see the link.

Cool