Prediction and Confidence Intervals

Plug $x_0$ into the fitted regression and you get a point estimate. The exam wants two intervals around that point, and they are not the same: one bands the average response, the other bands a single future outcome.

The fitted simple linear regression is $\hat{y}={\hat{\beta }}_0+{\hat{\beta }}_{1}x$. Plug in $x_0$ and the predicted value is ${\hat{y}}_0={\hat{\beta }}_0+{\hat{\beta }}_1{x}_0$. This single number serves two roles. As an estimate of the mean response $E[Y\mid X=x_0]$ it is unbiased. As a forecast for a single new observation $Y_0$ it is also unbiased. The point estimate is identical; only the uncertainty around it differs.

KEY: One point, two intervals. The narrow one (CI) covers where the average sits. The wide one (PI) covers where the next single observation could land.

A new observation decomposes as $Y_0={\beta }_0+{\beta }_1{x}_0+{\epsilon }_0$, with ${\epsilon }_0\sim N(0,{\sigma }^2)$. Estimating the mean response only requires nailing down ${\beta }_0+{\beta }_1{x}_0$; the noise ${\epsilon }_0$ is irrelevant because it averages to zero across many observations.

Common mistakes

• Using the CI formula when asked for a single new $Y_0$. Skipping the "+1" understates the band by roughly a factor of three to ten in typical setups. The exam stem uses words like "next," "individual," "a single," or "specific" for PI.
• Forgetting that PI uses the same $t_{n-p-1,\alpha /2}$ as the CI. The critical value does not change between intervals; only the standard error does. Candidates sometimes hunt for a different table.
• Computing $SE({\hat{y}}_0)$ without the leverage term. Dropping $(x_0-\bar{x}{)}^2/S_{xx}$ reduces to $s/\sqrt{n}$, which is the standard error of the mean of Y, not of the fitted value at a specific $x_0$. Wrong band, always too narrow at non-centroid X.

Bottom line

• A confidence interval bands the mean response $E[Y\mid X=x_0]$; a prediction interval bands a single new $Y_0$. The PI is always wider.
• The same point estimate ${\hat{y}}_0$ serves both intervals; only the standard error (uncertainty) differs.
• The PI standard error carries an extra "+1" inside the radical for irreducible noise ${\sigma }^2$. As $n\to \infty$ the CI collapses to a point but the PI keeps nonzero width.
• Both intervals widen as $x_0$ moves away from $\bar{x}$. They are narrowest at the centroid and bow outward via the leverage term $(x_0-\bar{x}{)}^2/S_{xx}$.

Exam shortcut

When the stem hands you $SE({\hat{y}}_0)$ directly, build the PI standard error as $\sqrt{SE({\hat{y}}_0{)}^2+s^2}$; the squared CI standard error plus $s^2$ equals the squared PI standard error. If you remember only one fact, remember that the PI radical has a "+1" and the CI radical does not. For 95% intervals with df above 30, swap $t\approx 1.96$ (or 2.0 rounded) without losing exam points.

The full lesson (about 3,345 words, 22 min read) adds 5 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 2f

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