My question is similar to this question, but the solution provided didn't tell whether increasing the sample size influences the prediction interval, so I would like to ask again.
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: $$ \hat y \pm t_{\alpha/2, n-2} \sqrt{MSE} \sqrt{1 + 1/n + \frac{(x-\bar x)^2}{\sum (x_i - \bar x)^2}} $$
If the sample size is increased, the standard error on the mean outcome given a new observation will decrease, then the confidence interval will become narrower. In my mind, at the same time, the prediction interval will also become narrower which is obvious from the fomular. However, my professor told me that the increasing sample size does not influence too much the prediction interval, so I am confused now. Could anybody give me some explanation?
self-study
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