Where is the prediction interval narrowest?

Topic: Linear regression · more questions on this topic
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In simple linear regression, at which value of x is the prediction interval for a new response narrowest, and what is its half-width there? Does it shrink to zero as n\to\infty?

Worked answer (click to show)

The half-width is t_{n-2;1-\alpha/2}\,S\sqrt{1+1/n+(x-\overline x)^2/S_{xx}}, which is smallest when (x-\overline x)^2=0, at x=\overline x. There it is t_{n-2;1-\alpha/2}\,S\sqrt{1+1/n}.

As n\to\infty, t_{n-2;1-\alpha/2}\to z_{1-\alpha/2}, S\to\sigma and 1/n\to0, so the half-width tends to z_{1-\alpha/2}\,\sigma, not to zero. However much data there is, a single new response still varies about the line with standard deviation \sigma.