Unit 7: Inference for Quantitative Data: Means

Confidence Interval for a Mean

Constructing and interpreting a one-sample t confidence interval for a mean.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Estimating a Mean When the Population Standard Deviation Is Unknown

Every confidence interval so far has assumed σ, the population standard deviation, is known. In practice it almost never is — only the sample's own standard deviation, s, is available. Before reading on: does estimating σ with s (an extra source of uncertainty) change how wide the resulting interval should be?

Definition — t-Distribution and Confidence Interval for a Mean

When σ is unknown (the usual case), the t-distribution replaces the Normal distribution: x̄ ± t*·(s/√n), with degrees of freedom df = n−1. The t-distribution is shorter and wider-tailed than the Normal — it accounts for the extra uncertainty of estimating σ from the sample itself, and it approaches the Normal distribution as df grows large. Conditions: random sample, 10% condition, and the population roughly Normal or the sample large enough (checked via a graph of the sample data) with no strong skew or outliers.

\bar{x} \pm t^* \frac{s}{\sqrt{n}}

Worked Example — Constructing a t Confidence Interval

A random sample of 25 batteries has x̄ = 18.2 hours, s = 3.5 hours (a graph of the sample shows no strong skew or outliers). df = 25−1 = 24; for 95% confidence, t* ≈ 2.064. SE = s/√n = 3.5/5 = 0.7. Margin of error = 2.064 × 0.7 ≈ 1.44. The interval: 18.2 ± 1.44 = (16.76, 19.64) hours.

Interpretation: 'We are 95% confident that the true mean lifetime of this type of battery is between 16.76 and 19.64 hours' — meaning that across many repeated random samples of this size, each producing its own confidence interval, about 95% of those intervals would capture the true population mean. As with a proportion's interval, this is not a 95% probability statement about this one already-computed interval containing the fixed true mean.

Tip

For any fixed confidence level, t* is always larger than the corresponding z* — the t-distribution's extra spread (from estimating σ) always produces a somewhat wider interval than the Normal-based formula would.

Common Mistakes

Using z* (1.96) instead of t* when the population standard deviation is unknown and s is used instead.
Whenever σ is unknown and estimated by s — nearly always in practice — the t-distribution and its correspondingly larger t* are required, not the Normal-based z*.
Using df = n instead of df = n−1.
One degree of freedom is used up estimating the mean itself from the sample, leaving n−1 — this is standard for a one-sample t-procedure.

Key Takeaways

When σ is unknown, a confidence interval for a mean uses the t-distribution with df = n−1, always producing a somewhat wider interval than the Normal-based formula.
Conditions require a random sample, the 10% condition, and either a Normal population or a large enough sample with no strong skew or outliers.
The interval's interpretation follows the same repeated-sampling logic as a proportion's confidence interval.

Summary

The next lesson uses this same t-distribution machinery to test a specific claimed value for a population mean.

Track your progress

Back to Unit 7: Inference for Quantitative Data: Means Significance Test for a Mean→