A confidence interval gives a range of plausible values for the population mean, built from the sample mean plus or minus a critical z-value times the standard error.

How to construct and correctly interpret a confidence interval for a population mean, the critical z-values for common confidence levels, and worked calculations.

Generated by Claude Opus 4.78 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The formula
Critical z-values
Interpreting the interval correctly
Common errors
Why it matters

What this dot point is asking

A single sample mean $\bar{x}$ is a point estimate of $\mu$ , but it is almost never exactly right. A confidence interval gives an honest range of plausible values for $\mu$ , with a stated level of confidence.

The formula

The interval is centred on $\bar{x}$ and extends a margin of error $z^*\dfrac{\sigma}{\sqrt{n}}$ either side.

Critical z-values

The critical value $z^*$ comes from the standard normal: it is the z-score leaving the right proportion in the central region.

Common critical values

Confidence level	$z^*$
$90\%$	$1.645$
$95\%$	$1.960$
$99\%$	$2.576$

A higher confidence level needs a larger $z^*$ , giving a wider interval.

Interpreting the interval correctly

The interpretation is precise and frequently examined:

Common errors

Why it matters

Confidence intervals are the headline inference technique of Topic 6 and a guaranteed exam item, valued both for the calculation and for the precise interpretation. They build directly on the standard error and Central Limit Theorem, and lead into the next dot point on margin of error and choosing a sample size.

Exam-style practice questions

Practice questions written in the style of SACE Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2017 SACE Stage 22 marksWool fibre diameter is normally distributed with a standard deviation of 2.5 microns. A producer randomly chooses 80 fibres and finds a sample mean of 17.4 microns. Calculate the 99% confidence interval for the population mean.

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Use the confidence interval for a mean: sample mean plus or minus z times (sd / sqrt(n)), with sd = 2.5, n = 80, and z = 2.576 for 99% confidence.

Standard error = 2.5 / sqrt(80) = 2.5 / 8.944 = 0.2795.

Margin of error = 2.576 times 0.2795 = 0.720.

Confidence interval = 17.4 plus or minus 0.720 = (16.7, 18.1) microns (to three significant figures).

Marks: one for the correct standard error and critical value (z = 2.576 for 99%), one for the assembled interval. A common slip is dividing the standard deviation by n instead of sqrt(n).

2023 SACE Stage 22 marksAfter a new feeding plan, a random sample of 20 cows has a mean daily milk production of 23.5 litres. Assuming the standard deviation is still 3.26 litres, calculate a 95% confidence interval for the mean daily milk production per cow on the new feeding plan.

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Apply the mean confidence interval with sample mean 23.5, sd = 3.26, n = 20, and z = 1.96 for 95% confidence.

Standard error = 3.26 / sqrt(20) = 3.26 / 4.472 = 0.729.

Margin of error = 1.96 times 0.729 = 1.43.

Confidence interval = 23.5 plus or minus 1.43 = (22.1, 24.9) litres (to three significant figures).

Marks: one for the correct standard error and z = 1.96, one for the final interval. This interval would then be compared with the old mean of 21.9 to judge whether the feeding plan increased production, but this part only asks for the interval itself.