 Solution

Let $F_{n-1}$ be the cumulative distribution function of the Student t-distribution. The two-sided p-value is $2[1-F_{n-1}(|t_0|)]$. If it is less than $\alpha$, strict monotonicity gives $|t_0|>t_{n-1,1-\alpha /2}$, henceThe magnitude of the interval's centre is larger than its half-width, so zero lies outside the confidence interval. This is the duality between confidence intervals and hypothesis tests.