Considering a simple OLS predictor $\hat\beta = (X^TX)^{-1}X^TY$, where $X$ is the design matrix and $Y$ is the response.
Given a new observation's covariate $x$, I can estimate the mean response using $x^T\hat\beta$, which follows a normal distribution with mean $x^T\beta$ and variance $\sigma^2x^T(X^TX)^{-1}x$. I understand the variance part determines the length of the confidence interval of the mean response with this specific new observation's covariate $x$.
I wonder what factor relates to this new observation's covariate $x$ affect the length. My immediate observation is that the larger the magnitude of $x$, the larger the variance, and therefore the larger the length.
But meanwhile my intuition talks about uncertainty - the more uncertain we are about a new observation, the high variance of the associated mean response, and thus the longer the length of CI.
So far it just seems the larger the magnitude of new observation $x$, the larger the uncertain we have about our estimated mean response, which does not make sense to me.
