Confidence interval vs prediction interval

Question

You fit a regression and want an interval at a new input x0. Distinguish a confidence interval for the mean response from a prediction interval for a new observation. Which is wider, and why?

Accepted Answer

The two answer different questions. Confidence interval (for the mean response). Bounds \mathbb{E}[y \mid x0] = x0^	op\beta, the average outcome at x0. Its only uncertainty is the estimation error in \hat\beta. In simple linear regression, \operatorname{Var}(\hat y0) = \sigma^2\left[\frac{1}{n} + \frac{(x0 - \bar x)^2}{S{xx}}ight], \qquad S{xx} = \sumi (xi - \bar x)^2. As n 	o \infty this variance 	o 0: with enough data you know the average outcome arbitrarily well. Prediction interval (for a new observation). Bounds a single future draw y0 = x0^	op\beta + \varepsilon0. This carries the…

Confidence interval vs prediction interval

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