If $B=\varnothing$, then for every $j$ at least one of $H_j^\beta$ and $H_j^\eta$ is true. Since
$$
\{q_j\leq t\}=\{q_j^\beta\leq t\}\cap\{q_j^\eta\leq t\},
$$
validity of the <p-value> for whichever component null is true implies $\mathbb P(q_j\leq t)\leq t$. Therefore the <union bound> gives
$$
\mathbb P\!\left(\min_{1\leq j\leq p}q_j\leq\frac\alpha p\right)
\leq\sum_{j=1}^p\mathbb P\!\left(q_j\leq\frac\alpha p\right)
\leq\alpha.
$$
This is the <Bonferroni correction> for the composite intersection alternatives.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 4 / c / Solution

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The Bonferroni correction tests each of $m$ hypotheses at level $\alpha/m$. The <union bound> then controls the <familywise error rate> by $\alpha$ under arbitrary dependence.

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 Incoming links: Bonferroni correction