# Do confidence intervals and prediction intervals shrink to a point for a very large sample size?

My question applies to regression estimates. The formulae for confidence interval: $$\hat y \pm t_{\alpha/2, n-2} \sqrt{MSE} \sqrt{1/n + \frac{(x-\bar x)^2}{\sum (x_i - \bar x)^2}}$$

and prediction interval: $$\hat y \pm t_{\alpha/2, n-2} \sqrt{MSE} \sqrt{1 + 1/n + \frac{(x-\bar x)^2}{\sum (x_i - \bar x)^2}}$$

both show that they decrease with increasing n and they both don't seem to tend to 0. But I remember vaguely someone telling me that CI shrinks to a point.

It would be great if someone could clarify this.

• A prediction interval better not shrink to a point! Otherwise there's no random behavior involved. – whuber Oct 12 '15 at 16:33
• Re the edit (which introduced the equations): it is evident that under mild conditions the confidence interval formula does tend to zero as $n$ grows, because (a) the $t$ term tends to the $1-\alpha/2$ quantile of the standard Normal distribution, which is finite, (b) $1/n$ tends to $0$, and (c) the other fraction in the square root tends to zero provided the $x_i$ don't all tend rapidly towards $\bar x$. – whuber Oct 12 '15 at 19:31