Two-stage sampling for acceptance testing

Sometimes a regulatory requirement or a quality-assurance procedure sets an allowed maximum on a confidence limit for a mean. If the sample mean of the measurements is below the allowed maximum, but the confidence limit is above it, a very widespread practice is to increase the sample size and recalculate the confidence bound. The confidence level of this two-stage procedure is rarely found correctly, but instead is typically taken to be the nominal confidence level, found as if the final sample size had been specified in advance. In typical settings, the correct nominal {alpha} should be between the desired P(Type I error) and half that value. This note gives tables for the correct a to use, some plots of power curves, and an example of correct two-stage sampling.