= Solution
Under $H_0$, each $Z_i=-\log X_i$ has <expected value> $1/2$ and <variance> $1/4$. The <central limit theorem> therefore approximates
$$
\frac{S-n/2}{\sqrt n/2}\ \mathrel{\dot\sim}\ N(0,1).
$$
For the supplied data this standardized value is
$$
\boxed{z_{\mathrm{obs}}=\frac{6.25-12.5}{2.5}=-2.5}.
$$
Using the provided approximation $\Phi(1.6)\approx0.95$, the lower-tail 5% rule rejects when $S\leq12.5-1.6(2.5)=8.5$. The observed $6.25$ is in this region, with approximate <p-value> $\Phi(-2.5)\approx0.005$. \b[Reject $H_0:\theta=2$ at the 5% level in favor of the specified alternative $\theta=3$.] The unusually large <geometric mean> is consistent with the direction of that alternative. This is evidence in this comparison, not a proof that the parameter is exactly three.
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