 Solution

Under $H_0$, each $Z_i=-\log X_i$ has expected value $1/2$ and variance $1/4$. The central limit theorem therefore approximatesFor the supplied data this standardized value isUsing the provided approximation $\Phi (1.6)\approx 0.95$, the lower-tail 5% rule rejects when $S\le 12.5-1.6(2.5)=8.5$. The observed $6.25$ is in this region, with approximate p-value $\Phi (-2.5)\approx 0.005$. 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.