# Statistical Hypothesis Testing for Asymmetric Tolerance Index  
[PERSON]  
1 Department of Industrial Engineering and Management, National Chin-Yi University of Technology, Taichung 41170, Taiwan; [EMAIL_ADDRESS] (K.-S.C.); [PERSON]_ADDRESS] (S.-C.C.); [EMAIL_ADDRESS] (W.-Z.C.)  
[PERSON]  
1 Department of Industrial Engineering and Management, National Chin-Yi University of Technology, Taichung 41170, Taiwan; [EMAIL_ADDRESS] (K.-S.C.); [PERSON]_ADDRESS] (S.-C.C.); [EMAIL_ADDRESS] (W.-Z.C.)  
[PERSON]  
3 Department of Business Administration, Chaoyang University of Technology, Taichung 413310, Taiwan 3*  
assumption, [PERSON] [17] proposed capability index \(C_{PK}\) on the basis of the process yield with symmetric tolerance:  
\[C_{PK}=Min\bigg{\{}\frac{USL-\mu}{3\sigma},\frac{\mu-LSL}{3\sigma}\bigg{\}}= \frac{1-|\delta^{\prime}|}{3\gamma^{\prime}}, \tag{1}\]  
where \(d^{\prime}=(USL-LSL)/2\), \(\delta^{\prime}=(\mu-T)/d^{\prime}\), and \(\gamma^{\prime}=\sigma/d^{\prime}\). According to [PERSON] [18], the relation between index \(C_{PK}\) and the process yield can be shown as follows:  
\[2\Phi\Big{(}3C_{pk}\Big{)}-1\ \leq\ Yield\%<\Phi\Big{(}3C_{pk}\Big{)} \tag{2}\]  
In Equation (2), \(\Phi(\cdot)\) refers to the function of cumulative distribution for the ordinary allocation. As mentioned above, the quality characteristics of many machinery products have asymmetric tolerances. A process has asymmetric tolerances when \(D_{u}=USL-T\) does not equal \(D_{l}=T-LSL\) (\(T\
eq M\)), where \(D_{u}\) is the upper tolerance and \(D_{l}\) is the lower tolerance. If \(d=Min\{D_{u},D_{l}\}\), \(d_{u}=d/D_{u}\), and \(d_{l}=d/D_{l}\), then the process capability index containing asymmetric tolerances suggested by [PERSON] and [PERSON] [19] can be defined as follows:  
\[C_{PK}^{\prime}=\frac{d-Max\{d(\mu-T)/D_{u},d(T-\mu)/D_{l}\}}{3\sigma}=\frac{ d-Max\{d_{u}(\mu-T),d_{l}(T-\mu)\}}{3\sigma}. \tag{3}\]  
Obviously, when \(T=M\) (a symmetric case), then \(C_{PK}^{\prime\prime}=C_{PK}\).  
Asymmetric tolerance index \(C_{PK}^{\prime\prime}\) is a function of \(\mu\) and \(\sigma\), which means it simultaneously depends on \(\mu\) and \(\sigma\). Unfortunately, the sampling distribution of index \(C_{PK}^{\prime\prime}\) is complex, making its \(100(1-\alpha)\%\) confidence interval hard to derive. As a result, this study adopts Boole's Inequality and DeMorgan's Theorem to find the combined confidence zone of \((\mu,\sigma)\). Next, using index \(C_{pk}^{\prime\prime}\) for the target function and the confidence zone of \((\mu,\sigma)\) for the feasible zone, this study applies mathematical programming to find the confidence interval of index \(C_{PK}^{\prime\prime}\) and employs this for statistical hypothesis testing capabilities of QCs with asymmetric tolerances meeting required quality levels. To sum up, the advantages of the model proposed in this study are that it (1) assesses the process quality of the asymmetric tolerance and symmetrical tolerance products; (2) uses mathematic programming to deduce confidence interval easily without regarding the complicated distribution of the estimated formula of the estimator of index \(C_{PK}^{\prime\prime}\); (3) reduces the misjudgment risk of sampling error owing to testing through the confidence interval.  
The other sections of this study are arranged as follows: Section 2 displays the \(100(1-\alpha)\%\) confidence interval of the asymmetric tolerance index \(C_{PK}^{\prime\prime}\), Section 3 demonstrates the \(100(1-\alpha)\%\) confidence interval used to construct statistical testing rules, Section 4 provides an illustrative example to explain the applicability of the method proposed by this study, and Section 5 presents conclusions. A conceptual framework of the Introduction section is shown in Figure 1.