Confidence intervals for the mean

In short

• A point estimate (e.g. $\bar X$) is a single best guess of a parameter. An interval estimate, or confidence interval, is a range built around it with a stated confidence level $1 - \alpha$.
• Structure: point estimate ± reliability factor × standard error. The reliability factor is the critical value (z or t) that leaves $\alpha/2$ in each tail.
• z-values to know: 1.645 (90%), 1.96 (95%), 2.576 (99%). A value with 0.025 in one tail belongs to a 95% interval; 0.05 in one tail belongs to a 90% interval.
• Use z when $\sigma$ is known (normal population, or any population with a large sample). Use t with n − 1 degrees of freedom when $\sigma$ is unknown. With a small sample from a clearly non-normal population, neither z nor t gives a valid interval.
• The interval widens with a higher confidence level, a larger standard deviation, or a t instead of a z. It narrows as n grows, in proportion to $1/\sqrt{n}$.