Confidence Intervals for Means

Key idea

Range likely to contain the true mean.

In depth

An interval x̄ ± t*·(s/√n) captures the true mean in 95% (or the chosen level) of repeated samples. It reflects uncertainty from sampling, with width shrinking as n grows.

Worked example

x̄ = 50, s = 10, n = 25, t* ≈ 2.06: 50 ± 4.1 → (45.9, 54.1).

Formula

Real-world uses

• Reporting the likely range of average heights or average response times in a study

Common mistake

Interpreting it as a 95% chance the mean lies in this one interval.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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