Prediction Intervals

A prediction interval is simply a range where the next single part off the line will probably land, as opposed to a range for the average of all parts. The formula $\bar x \pm t\,s\sqrt{1 + 1/n}$ is the familiar confidence interval with one change under the square root: the $1$ (the new part's own scatter, which never goes away) plus the $1/n$ (our uncertainty about where the mean is, which shrinks with more data). That little "1 +" is the whole difference, and it turns out to matter a lot.

One part, not the average. A customer doesn't install your process mean; they install the next bracket off the line. "How strong will the next one be?" is a different question from "what's the mean strength?", and it needs a different interval.

Derivation. Predict the new part $X_{\text{new}}$ by $\bar X$. The new part is independent of the sample, so the prediction error has mean 0 and variance

because variances of independent pieces add even when you subtract, the same rule behind tolerance stack-up. Standardize, then replace $\sigma$ with $s$. For normal data $s$ is independent of both $\bar X$ and $X_{\text{new}}$, so

Trap $T$ between $\pm t_{\alpha/2,n-1}$ and solve for $X_{\text{new}}$.

Why it's wider, and stays wide. The half-widths are $t\,s/\sqrt n$ (confidence) and $t\,s\sqrt{1 + 1/n}$ (prediction), a ratio of $\sqrt{n + 1}$. As $n$ grows the confidence interval shrinks to a point, but the prediction interval only shrinks to $\mu \pm z_{\alpha/2}\sigma$. Data make you surer of the mean; they never make the parts less variable. Only a better process does that.

One-sided bounds. For a minimum strength, use $\bar x - t_{\alpha,n-1}s\sqrt{1 + 1/n}$, with all of α in one tail.

What "95%" means here. Over repeated rounds of (sample, then one more part), the interval catches that part 95% of the time. It does not promise that 95% of all future parts fall inside; that's a different question, answered by a tolerance interval.

Normality matters more. The CLT rescues $\bar x$ from skewed data, but the new part is a single observation: if the process is skewed, so is the prediction error. Check a probability plot.

Connections: the interval it extends is in confidence intervals for a mean, and regression adds the same "1 +" term when it predicts a new reading from a calibration curve.

Strength allowables, battery and LED datasheets and sensor calibrations all ask about the next unit, so they use prediction intervals. The same interval flags anomalies: build a 99% prediction interval from the last 30 good parts, and a new part outside it deserves a look before it ships. That idea, repeated over time, is the logic behind control charts.