Lesson 7 of 8 · 13 min

Forecasts and prediction intervals

A forecast is a point on the fitted line; the prediction interval around it widens with the SEE, with a small sample and with how far the forecast X is from the mean of X.

In short

Point forecast: $\hat Y_f = \hat b_0 + \hat b_1X_f$ .
The standard error of the forecast $s_f$ is always larger than $s_e$ : it adds uncertainty about the estimated line itself.
$s_f$ is smaller with a better fit (lower $s_e$ ), a larger n, more variation in X, and an $X_f$ closer to $\bar X$ .
Prediction interval: $\hat Y_f \pm t_c \times s_f$ , with n − 2 df and $t_c$ for $\alpha/2$ in each tail.
A higher confidence level gives a larger $t_c$ and a wider interval.

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