How Do You Interpret a Confidence Interval in Context?

A confidence interval should be interpreted by naming the population parameter and stating that you are confident it lies between the calculated bounds. For example: “We are $95\%$ confident that the population mean weekly study time lies between $12.4$ and $15.1$ hours.”

What the Confidence Level Means

Suppose repeated random samples were taken and a $95\%$ confidence interval was calculated from each sample. In the long run, approximately $95\%$ of those intervals would contain the true population parameter, such as the population mean $\mu$.

Once an interval has been calculated, the unknown parameter is fixed: it either lies inside that particular interval or it does not. The confidence refers to the reliability of the interval-generating method, not to a changing probability for $\mu$.

A confidence interval estimates a population parameter, not the spread of individual data values. The sample mean is already known from the sample, so the interval is not estimating it.

For AHL 4.16, use the normal distribution when the population standard deviation $\sigma$ is known and the $t$-distribution when $\sigma$ is unknown, regardless of sample size.

Exam Technique

For the command term Interpret, include the confidence level, both bounds, the population parameter, units, and context. Avoid the common misconception that a $95\%$ confidence interval contains $95\%$ of the data or gives a $95\%$ probability about a fixed parameter. This is HL-only content and may be tested in GDC-active Paper 1 or Paper 2 questions.