Confidence Intervals

A range of plausible parameter values, with a stated coverage rate.

A confidence interval is not a probability statement about the parameter — it's a statement about the procedure that produced it. A $95\%$ confidence interval is one that, in repeated sampling, contains the true parameter $95\%$ of the time.

For a sample mean from a Normal (or large $n$ via CLT), the standard $95\%$ interval is

\bar{X} \pm z_{\alpha/2}\, \frac{s}{\sqrt{n}}, \qquad z_{0.025} \approx 1.96

A common and important misinterpretation: a $95\%$ CI does not mean "the parameter is in this interval with probability $0.95$ ." That's the Bayesian credible interval, which requires a prior. The frequentist interval makes a statement about the procedure, not the realized interval. Once you've computed it, the parameter is either in it or not — there's no probability at the realized level.