Confidence intervals
A confidence interval is a range for a parameter, built so that a stated fraction of such intervals (across many samples) cover the truth.
For large enough $n$, a handy 95% CI for the population mean is “sample mean ± about two standard errors”.
$$m_x \pm 1.96\,\frac{s_x}{\sqrt{n}}$$
Reading a 95% CI
Suppose $m_x = 39$ minutes and SE $= 3.3$, so the interval is roughly $39 \pm 6.5$, i.e. about 32.5 to 45.5. Across many random samples of the same size, about 95% of such intervals would contain the true mean commute. Your one interval either does or does not — you cannot know which.
Assumptions
This normal-based interval is approximate: typically want $n \ge 30$ and the sample a small fraction of the population. A 100% CI from $-\infty$ to $\infty$ is useless — we trade coverage for a useful width.
Key takeaways
- CI = interval estimate + confidence level.
- 95% ≈ $\pm 1.96$ SE for means (large $n$).
- Coverage is a long-run property of the procedure.