duality between hypothesis tests and CIs

Hypothesis tests can be inverted to create confidence intervals and, conversely, confidence intervals can be used to derive hypothesis tests.

The intuition is straightforward: A hypothesis test of a point null tests whether a parameter ${\theta }^{\ast }$ is equal to a particular value ${\theta }_0$. If we gather all ${\theta }_0$ such that the null is not rejected, then this is a confidence interval. Likewise, if we have a CI $C$, we can simply reject the null $H_0:{\theta }^{\ast }={\theta }_0$ if ${\theta }_0\notin C$.

This duality also holds for confidence sequences, see eg my notes here.