Test claim using a confidence interval (a, b)

Title

Test claim about $\mu$ using a confidence interval $(a,b)$

• $\mu ={\mu }_0$: Do not reject if ${\mu }_0\in (a,b)$, reject otherwise
• $\mu <{\mu }_0$: Do not reject if $b<{\mu }_0$, reject otherwise
• $\mu >{\mu }_0$: Do not reject if ${\mu }_0<a$, reject otherwise
• SAME FOR $p$