= Solution
Three defensible approaches are:
* Use simultaneous calibration, for example <Bonferroni correction>: give each of eight hospitals outside <probability> at most $0.05/8$. Replace 1.96 by $z_{1-0.05/16}\approx2.7344$ in the normal bands, giving familywise error at most 5% when the marginal tests are calibrated.
* Apply the <Holm step-down procedure> to valid two-sided binomial p-values. Order them and compare the jth smallest with $0.05/(9-j)$ until the first nonrejection. This controls familywise error while often being less conservative than giving every hospital the same Bonferroni threshold.
* Treat an initial signal as a screen and require prespecified confirmation in independent new data before declaring a persistent deviation, alongside checking case mix and coding. Calibration must include both stages; unrestricted repeated testing until an alarm appears would increase, rather than solve, the problem.
\b[A control-limit crossing prompts investigation; it is not itself a verdict on a hospital.] Wider or adjusted thresholds also reduce sensitivity, so the false-alarm objective and the cost of missed genuine problems should be made explicit.
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