AHL 4.16 (HL)—Confidence intervals

A confidence interval combines a sample estimate with a margin of error to give a range of plausible values for a population parameter under a stated procedure and confidence level.

The interval widens when variability increases or the sample becomes smaller, and narrows when the sample is larger. Its interpretation is about the long-run performance of the method, not a probability assigned to a fixed parameter after the interval is calculated.

If a mean estimate is 72 with margin 4, a 95% interval is (68,76) under the model used. Repeating the sampling procedure would produce intervals that capture the true mean about 95% of the time in the long run.

A confidence level is not the chance that this already-fixed interval contains the parameter. Check assumptions, units and whether the target parameter matches the estimate.

For a normal population mean, use $\bar x\pm z^*\sigma/\sqrt n$ when population $\sigma$ is known and $\bar x\pm t^*s/\sqrt n$ with $n-1$ degrees of freedom when $\sigma$ is unknown, regardless of sample size. Report the confidence level, parameter, units and population in the contextual interpretation.