= Solution
Set $h=\mathbb E\theta>0$ and $w=\operatorname{Var}(\theta)>0$. Keeping $h$ fixed in the <gamma distribution> family gives $\alpha=h^2/w$ and $\lambda=h/w$. The ratio of <expected process variance> to <variance of hypothetical means> becomes
$$
K=\frac{h^2+w}{w(k-2)}=\frac{1+h^2/w}{k-2},\qquad
Z=\frac{n(k-2)}{n(k-2)+1+h^2/w}.
$$
As the prior <variance> $w$ decreases, $K$ increases, so \b[the <credibility factor> decreases]. In the limit $w\downarrow0$, $Z\downarrow0$: the risk parameter is already essentially known from the prior, leaving little need to use the noisy <sample mean>. The population claim <expected value> $h/(k-1)$ remains fixed throughout this comparison.
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