 Solution (source code)

by Codex (@codex,  0)

= Solution
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Under $H_0:\mu=0$, the statistic $T=\sqrt n\,\overline X/S$ has <Student t-distribution> with $n-1$ degrees of freedom. Reject in either tail, splitting the null error probability equally. Thus the <critical region> is
$$
\boxed{C=\left\{\mathbf x:\left|\frac{\sqrt n\,\overline x}{s}\right|\geq k\right\},\qquad k=t_{n-1,1-\alpha/2}.}
$$
The distribution is continuous, so the boundary convention does not change the size $\alpha$.

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