Confidence intervals: meaning and interpretation

A confidence interval is an estimate ± margin of error produced by a procedure that captures the true parameter in a stated proportion of repeated samples.

\text{estimate} \pm \text{critical value} \times \SE

General form of most intervals

$\text{estimate}$: the point estimate, e.g. $\xbar$ or $\phat$
$\text{critical value}$: a quantile such as $z_{1-\alpha/2}$ or $t_{1-\alpha/2,\,df}$
$\SE$: standard error of the estimate
$1-\alpha$: confidence level, e.g. 0.95

When to use

Whenever you report an estimate: an interval shows the precision as well as the value. Prefer reporting an interval alongside (or instead of) a bare p-value.

Assumptions

Those of the specific interval (random sampling, independence, approximate normality of the estimate). The margin of error covers random sampling error only, not bias from the design or measurement.

How to interpret

Correct: "We are 95% confident that the mean lies between 12.0 and 15.0", meaning the method captures the true value in 95% of repeated samples.
After the data are in, a particular interval either contains the fixed parameter or it does not; the 95% describes the procedure's long-run success rate, not a probability about this interval.
Wider intervals: higher confidence, more variable data, smaller $n$ .
A 95% two-sided interval excludes a value $\theta_0$ exactly when a two-sided test at $\alpha = 0.05$ (of the matching type) rejects $\theta_0$ .
For a probability statement about the parameter itself, use a Bayesian credible interval.

Common mistakes

"There is a 95% probability that μ is in (12.0, 15.0)."
"95% of the data lie in the interval." The interval is about a parameter, not individual values (that is a prediction interval).
Treating overlapping intervals for two groups as proof of no difference; compute an interval for the difference.

Example

From $\xbar = 13.5$ , $s = 1.7728$ , $n = 8$ : the 95% t interval is $13.5 \pm 2.3646 \times 0.6268 = (12.02, 14.98)$ . Repeating the study many times, about 95% of such intervals would capture the population mean.