Construct and interpret confidence intervals for a population mean using the sample mean and standard error, choosing the appropriate confidence level, and understand the meaning of the confidence level in repeated sampling

What this dot point is asking

QCAA wants you to build an interval estimate for an unknown population mean from a sample, choose a confidence level, compute the margin of error, and interpret the interval correctly in repeated-sampling language. Confidence intervals are a major theme of IA3 and the external assessment, and the interpretation is examined as carefully as the calculation.

The answer

Point estimate versus interval estimate

A single sample mean $\bar{x}$ is a point estimate of the population mean $\mu$, but it almost certainly is not exactly $\mu$. A confidence interval gives a range of plausible values for $\mu$, built around $\bar{x}$, together with a stated level of confidence.

Structure of the interval

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

where $\dfrac{\sigma}{\sqrt{n}}$ is the standard error and $z^{*}$ is the critical value from the standard normal distribution for the chosen confidence level. The quantity $z^{*}\dfrac{\sigma}{\sqrt{n}}$ is the margin of error, the half-width of the interval.

Critical values

The critical value $z^{*}$ cuts off the central proportion of the standard normal equal to the confidence level. The standard values are:

A higher confidence level needs a larger $z^{*}$, so the interval is wider. There is a trade-off: more confidence costs precision.

Width, sample size and confidence level

The interval width is $2z^{*}\dfrac{\sigma}{\sqrt{n}}$. It widens with a higher confidence level (larger $z^{*}$) and narrows with a larger sample (the $\sqrt{n}$ in the denominator). To halve the margin of error you must quadruple the sample size, because of the square root.

Interpreting the confidence level

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.

Using a sample standard deviation

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.

Recovering missing information

Examiner-favoured problems give you an interval and ask you to work backwards. Because the half-width equals $z^{*}\dfrac{s}{\sqrt{n}}$, knowing any three of the four quantities (half-width, $z^{*}$, $s$, $n$) lets you solve for the fourth. A common task is to read the half-width as half the difference of the endpoints, then rearrange to find the sample size $n$ or the sample standard deviation $s$. Once $n$ is recovered, you can rebuild the interval at a different confidence level using the same $\bar{x}$ and $s$, as the elephant-mass question above shows.

Making a decision from an interval

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.