Process capability indices Cp, Cpk, and Cpm are still nowadays widely used in industry–thanks to their easy formulation and implementation. This paper aims to give new mathematical insights in order to support their use in decision‐making via hypothesis testing. The minimum sample size usually needed in the applications to achieve fixed significance level and power is reported in light of the new mathematical aspects for Cpk and Cpm, which avoid misleading conclusions and the use of extensive numerical experiments. In addition, power curves for Cpk and Cpm, which have not previously appeared in the literature before, are also presented. Lastly, easy‐to‐follow diagrams for hypothesis testing with Cpk and Cpm and two critical scenarios for Cpm are included in the paper to facilitate the applicative use and the comprehension of the novel inferential aspects.
