Confidence vs prediction intervals in regression

A confidence interval estimates the mean response at x₀; a prediction interval covers a single new observation and is always wider.

\hat y_0 \pm t_{1-\alpha/2,\,n-p}\,\hat\sigma\sqrt{\mathbf{x}_0^\top (X^\top X)^{-1}\mathbf{x}_0}

Confidence interval for the mean response

$\hat y_0$: fitted value at the new predictor values
$\mathbf{x}_0$: row of predictor values (with a 1 for the intercept)
$X$: model matrix of the fitted data

\hat y_0 \pm t_{1-\alpha/2,\,n-p}\,\hat\sigma\sqrt{1 + \mathbf{x}_0^\top (X^\top X)^{-1}\mathbf{x}_0}

Prediction interval for one new observation

$1$: the extra term for the new observation's own error variance

\mathbf{x}_0^\top (X^\top X)^{-1}\mathbf{x}_0 = \frac{1}{n} + \frac{(x_0-\xbar)^2}{S_{xx}}

Simple regression form

$x_0$: the predictor value at which you predict

When to use

Confidence interval: "What is the average y for all units with x = x₀?"
Prediction interval: "Where will the next individual with x = x₀ fall?"

Assumptions

Those of regression inference. Prediction intervals additionally rely heavily on normal, constant-variance errors — they do not improve with n the way confidence intervals do.

How to interpret

Both are narrowest at $x_0 = \xbar$ and widen as $x_0$ moves away.
As n grows the confidence interval shrinks towards zero width; the prediction interval shrinks towards $\hat y_0 \pm z\sigma$ , never to zero.

Common mistakes

Using the confidence interval to describe where individual observations will fall (far too narrow).
Predicting outside the range of the data, where neither interval is trustworthy.

Example

Spend–sales line at x₀ = 4: $\hat y_0 = 6.6$ , t(4 df) = 2.776.

Mean response: SE = 0.356, 95% CI (5.61, 7.59).
One new week: SE = 0.909, 95% PI (4.08, 9.12) — about 2.5 times wider.