Duality between confidence intervals and hypothesis tests ($C(x)=\{\theta :H_{\theta }\text{ is accepted at }x\}$)

Invert a family of hypothesis tests by including each parameter value in the confidence set exactly when its test does not reject. For any true value $\theta$, the events $\{\theta \notin C(X)\}$ and $\{\text{reject }{H}_{\theta }\}$ are identical. Thus size at most $\alpha$ gives coverage at least $1-\alpha$, and exact size gives exact coverage. When the accepted set is an interval, this is a confidence interval.

For the normal sample mean with unknown variance, the central Student t-distribution interval excludes zero exactly when $|\sqrt{n}\overline{X}/S|$ exceeds the same two-sided critical quantile. A two-sided p-value strictly below $\alpha$ therefore implies exclusion of zero from the matching closed interval. Boundary conventions agree up to events of probability zero in this continuous model.