Confidence intervals for a population mean with known variance, the meaning of a confidence level, the effect of sample size and confidence level on width, and using the t distribution when the variance is unknown.

What this dot point is asking

AQA wants you to construct a confidence interval for a population mean when the variance is known, interpret the confidence level correctly, explain how sample size and confidence level affect the width, and use the t distribution to build an interval when the population variance is unknown and estimated from the sample.

Confidence interval with known variance

Interpretation

The distinction matters because once the interval is computed it either contains the true mean or it does not, so it is wrong to say there is a $95\%$ probability the mean lies inside this specific interval. The probability statement attaches to the procedure before the sample is drawn. Examiners reward the careful wording: it is the intervals, varying from sample to sample, that contain the fixed unknown mean $95\%$ of the time.

Width, sample size and the t distribution

The width of the interval is controlled by two things. A higher confidence level uses a larger critical value, so the interval widens, which is the price of greater certainty. A larger sample increases $\sqrt{n}$ in the denominator of the standard error, so the interval narrows; to halve the width you must quadruple the sample size, because the dependence is on $\sqrt{n}$ rather than $n$.

When the population variance is unknown, you estimate it from the sample using the unbiased estimator $s^2$ and replace $z$ by a value from the t distribution with $n - 1$ degrees of freedom. The t distribution has heavier tails than the normal, reflecting the extra uncertainty in estimating $\sigma$, so its critical values are larger and the interval is correspondingly wider. As the sample grows the t distribution approaches the normal, so for large samples the two approaches give almost the same interval, but for small samples the difference is substantial and using $z$ would understate the uncertainty.

A point examiners test is the link between confidence intervals and hypothesis testing. A two-sided $95\%$ confidence interval contains exactly the values of the mean that would not be rejected by a two-tailed test at the $5\%$ significance level. So if a claimed mean lies outside the interval, a test at the matching level would reject it, and if it lies inside, the test would not reject it. This duality lets you answer a hypothesis question directly from an interval, and it explains why a wider interval (higher confidence) is harder to fall outside, just as a smaller significance level is harder to reject at. Keeping the confidence level and the test's significance level paired (a $99\%$ interval matching a $1\%$ test) is the key to using one to answer the other.