Statistics · step 8 of 13

Confidence intervals for a mean

A confidence interval is x̄ ± (critical value) × (standard error). Use z when σ is known, and t with n − 1 degrees of freedom when σ is estimated from the data — which is nearly always. The interpretation is subtle and heavily examined: 95% refers to the long-run success rate of the PROCEDURE, not the probability that this particular interval contains μ.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Choose z or tσ known (rare) means z; σ estimated by s means t with df = n − 1.
Find the critical value1.96 for a 95% z-interval; for t, read the table at your df and confidence level.
Compute the margin of errorcritical value × s/√n. This is the half-width of the interval.
State the interval and interpret it in context"We are 95% confident the population mean lies between … and …", referring to the population, not the sample.

Worked example

A sample of n = 25 has x̄ = 68 and s = 10. Build a 95% confidence interval for μ.

σ is unknown, so use t with df = 24. The critical value is t₀.₀₂₅,₂₄ ≈ 2.064.
Standard error = s/√n = 10/5 = 2.
Margin of error = 2.064 × 2 = 4.13.
Interval: 68 ± 4.13.

Answer. (63.87, 72.13). Using z = 1.96 instead would have given (64.08, 71.92) — too narrow, because it ignores the uncertainty in estimating σ.

Where marks get dropped

These are the specific errors that cost credit on confidence intervals for a mean questions — QED's rubric penalises each of them separately.

Saying "there is a 95% probability μ is in this interval". μ is fixed; the interval is random. The 95% describes the procedure over repeated samples.
Using z when σ was estimated from the sample. For small n the t critical value is noticeably larger.
Interpreting the interval as containing 95% of the DATA. It is an interval for the mean, and it is far narrower than the data range.

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Confidence intervals for a mean — frequently asked questions

What does 95% confidence actually mean?

If you repeated the whole sampling procedure many times, about 95% of the intervals produced would contain the true μ. Any single interval either does or does not.

How do I halve the margin of error?

Quadruple the sample size, since the margin scales as 1/√n. Alternatively accept a lower confidence level.

When is z acceptable instead of t?

When σ is genuinely known, or when n is large — beyond about 30 the t and z critical values differ by under 2%.

The rest of Statistics

Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.

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