= Solution
Under the point hypothesis $\beta=0$, the <test statistic> $\sqrt n\,\overline Y$ has a standard <normal distribution>. The observed statistic is three, giving a two-sided <p-value> $2[1-\Phi(3)]\simeq0.00270$. This is conventionally strong evidence against that point hypothesis. The narrow model $H_0$ is a continuous <prior distribution> around zero, rather than a point hypothesis.
For $n_0/n=100$ and $c=100$, <Normal-normal conjugacy> gives
$$
\boxed{\beta\mid y,H_0\sim N\left(\frac{3}{101\sqrt n},\frac1{101n}\right),\qquad
\beta\mid y,H_1\sim N\left(\frac{300}{101\sqrt n},\frac{100}{101n}\right).}
$$
The narrow-model <posterior mean> is strongly pulled toward zero and its <standard deviation> is about $0.0995/\sqrt n$. The wide-model <posterior mean> is about $2.9703/\sqrt n$, very close to the observation, with <standard deviation> about $0.9950/\sqrt n$. \b[Each model produces a markedly different posterior], so choosing between them requires their predictive evidence.
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