Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 37 / 4 / v

The source plot's sharp rise in the final year crosses the chosen control boundary. Trigger a prompt independent investigation of the mortality signal. First verify records, deaths, follow-up and expected-count calculations, then review case mix, changes in referral or recording, and plausible clinical explanations. The response should include appropriate clinical governance and safeguards while the cause is assessed; a statistical signal is not, by itself, proof of misconduct or a particular cause.

The stated null average run length of $111$ years makes a signal unusual under the calibrated null, while the alternative value of $5.2$ years indicates sensitivity to a sustained doubling under the specified model. Neither number gives the posterior probability that a doubling has occurred, and $1/111$ is not automatically the false-alarm probability for this particular ten-year history. Likewise the threshold is not a standalone fixed-test significance level.

The earlier value $X_{1993}=0$ says only that, at that point, no recent accumulated positive log-likelihood ratio survived the resets. It is not evidence that the earlier death rate was certainly normal, and does not cancel the later signal. The zero reset is deliberately part of the change-detection design; the 1994–1996 increments can independently justify an alarm. The full earlier clinical history remains relevant to investigation, even though a negative earlier accumulated score is not carried into the new CUSUM segment.