Quick questions on Confidence intervals for a population mean (QCE Specialist Mathematics Unit 4)

A confidence interval for $\mu$ (with known population standard deviation $\sigma$) is

The correct interpretation refers to the method over repeated sampling. A $95\%$ confidence level means that if many samples were taken and an interval constructed from each, about $95\%$ of those intervals would contain the true mean $\mu$. It does not mean there is a $95\%$ probability that $\mu$ lies in this particular interval: $\mu$ is a fixed number, and any single interval either contains it or does not. The confidence is in the long-run reliability of the procedure.

In strict theory the formula uses the known population standard deviation $\sigma$. In practice $\sigma$ is rarely known, so QCE uses the sample standard deviation $s$ in its place, writing the interval as $\bar{x} \pm z^{}\dfrac{s}{\sqrt{n}}$. This is justified by the central limit theorem and the large samples used in assessment items: for the sample sizes QCAA sets, the normal critical values $z^{}$ remain a good approximation, which is why the syllabus describes these as approximate confidence intervals. The standard error $\dfrac{s}{\sqrt{n}}$ is the single most important quantity to compute correctly, since every other step depends on it.

A confidence interval doubles as an informal hypothesis test. If a claimed value of the mean lies inside the interval, the data are consistent with the claim at that confidence level; if it lies outside, there is evidence against the claim. Phrasing matters: write that the value is or is not contained in the interval, and state the confidence level, rather than declaring the claim simply true or false. Because higher confidence widens the interval, a value can sit outside a $95\%$ interval yet inside a $99\%$ interval, so always tie the conclusion to the stated level.