Authors’ response

We welcome the critique from Dr Wellek and Dr Jennen-Steinmetz. In particular, we share their concerns that a reference range (RR) should not be unnecessarily wide, leading to an avoidable low sensitivity of the RR. When a $P$ -content and $\gamma$ -confidence tolerance range (TR) is used as RR, this concern can be addressed by simply increasing the sample size $n$ to improve its sensitivity since ‘as the sample size increases, the computed tolerance interval will approach the probability interval that actually contains the specified population proportion’. ¹ ${}^{(\mathrm{p}.150)}$ To be more precise, it has been proposed ² to choose the sample size $n$ to control the sensitivity of a $(P,\gamma )$ TR in terms of $$Pr\{\mathrm{TR}\mathrm{contains}\mathrm{at}\mathrm{least}100P^{*}\text{\%}\mathrm{of}\mathrm{the}\mathrm{population}\}\le {\delta }_0$$ where $P<P^{*}<1$ is pre-specified for a sufficiently small value of ${\delta }_0>0$ . This sample size formula guarantees that there is no more than a small chance ${\delta }_0$ that the TR includes a proportion of $100P^{*}\text{\%}$ or more of the population. Tabulation of the sample sizes for the normal distribution can be found in Faulkenberry and Daly ³ and Hahn and Meeker ¹ ${}^{(\mathrm{Ch}.9)}$ or easily computed by using the R package tolerance. ⁴