Precision follows the decision: how well an acceptance probability is known for mean-spread release rules

Abstract

Many release rules accept a lot when a statistic built from the sample mean and standard deviation stays within a limit, and the probability that a process passes such a rule can be computed exactly for a broad class of parent distributions. This paper asks how well that probability is itself known once the parent has been estimated. Under a stated reference posterior the answer is an interval, narrow where a lot clearly passes or fails and wide where it is contested. In ninety-eight published content-uniformity lots that position, not the sample size, sets the width: among the forty-eight lots assayed at the same ten units, and so equally informed about the spread, the $95\%$ interval runs from essentially zero width to $0.96$. With the operating point held fixed while test and estimation sample grow together, the interval narrows only slowly, the contraction of the posterior being offset by the steepening of the operating characteristic; a calibrated study at the sample sizes of $1207$ occupational exposure assessments shows the same under a half-plane rule. For a pure-spread rule under a Normal parent the interval has an exact chi-square form in the plug-in value and the two sample sizes, extending to the mean-spread wedge with a measured residual and containing Bergum's lower bound as one endpoint. With the parent's posterior formed from the same published summaries through the evaluator's own likelihood, the qualitative pattern survives but shape moves both endpoints and width: on the same lots a strong right skew lowers the pessimistic endpoint by up to a fifth and widens the interval by up to $0.22$. At the content-uniformity operating point studied, under a Normal parent, the symmetric error bar in standard use puts the lower endpoint $0.21$ too high and returns an upper endpoint above one, even with the full posterior covariance.