Prediction Intervals

Prediction intervals estimate the range within which a future observation from the same distribution is likely to fall, with a specified confidence level.

Formula:

$\bar{x}\pm t_{\alpha /2,n-1}\cdot s\cdot \sqrt{1+\frac{1}{n}}$

Where:

• $\bar{x}$: Sample mean
• $s$: Sample standard deviation
• $n$: Sample size
• $t_{\alpha /2,n-1}$: t-critical value for the chosen confidence level

Notes:

• Prediction intervals are always wider than a confidence interval for the mean.
• They use the t-distribution due to sample uncertainty.
• The interval is centered around $\bar{x}$ but accounts for:
  • Estimation error of the mean
  • Natural variability of new values

Use Cases:

• Forecasting where a new measurement is likely to fall.
• Risk assessment and operational thresholds.