Confidence Interval

A confidence interval (CI) is a range of plausible values for a population parameter (e.g. mean, proportion), constructed from sample data with a stated confidence level (commonly 95%).

For a population mean with known $\sigma$, the 95% CI is

$\bar{x} \pm 1.96 \cdot \frac{\sigma}{\sqrt{n}}$

where $1.96$ is the 97.5th percentile of the standard normal (corresponds to 95%).

Correct interpretation: "If we repeated this sampling procedure many times and built a CI each time, about 95% of those CIs would contain the true parameter." It is a statement about the procedure's long-run reliability, not about the specific interval.

Common misinterpretation (drilled by every stats teacher): "There's a 95% probability the true value is in this specific interval." Wrong — the parameter is fixed; the interval is random.

The confidence level controls a tradeoff:

• 99% CI: more confident, wider interval.
• 90% CI: narrower, less confident.

CIs are the modern alternative to p-values: they convey the same information about statistical significance plus the magnitude of the effect.