Statistical considerations for using tolerance interval to set product specification for normally distributed attribute

Conventionally, the product quality specification and control chart limits are determined as the mean plus and minus 3 sample standard deviations with the assumption that the quality data is normally distributed. These limits correspond to an interval centered at the mean, covering approximately 97.3% of the population. The estimate of such an interval is called the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-content tolerance interval. It has been proposed to use a two one-sided <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-content tolerance interval approach for determining drug product quality specifications. For a given confidence level, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo></mml:math> and a coverage percentage <i>p</i>, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-content tolerance interval is not precise when the sample size is small. For the derivation of a precise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-content tolerance interval, Faulkenberry and Daly proposed a "goodness" criterion for sample size determination. In order to avoid overestimating the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-content tolerance interval when <i>p</i> is large, we propose to define the precision requirement as the probability of the tolerance interval covering more than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mfrac><mml:mrow><mml:mfenced><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math> is restricted to a pre-specified significance level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>α</mml:mi><mml:mo>'</mml:mo></mml:msup></mml:math>. Quality specification studies are often not planned with proper sample sizes. To obtain precise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-content tolerance intervals for quality specification studies, the proper coverage <i>p</i> satisfying the "goodness" criterion and the minimum sample sizes were also determined with the pre-specified significance level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>α</mml:mi><mml:mo>'</mml:mo></mml:msup></mml:math>. With this approach, one may properly set the product specificationwhile avoiding over-specifying the quality limits.