# Statistical Hypothesis Testing for Asymmetric Tolerance Index
## 3 Statistical Hypothesis Testing  
As described in the previous section, the \(100(1-\alpha)\%\) confidence interval of the asymmetric tolerance index \(C^{\prime\prime}_{pk}\) is represented by the following specification limits:  
\[LC^{\prime\prime}_{pk}=\left\{ \begin{array}{l}\hat{C}^{\prime\prime}_{pk0} \times\sqrt{\frac{\chi_{2/4\mu-1}^{2}}{n-1}},\ T\in[\overline{x}-e_{t0},\ \overline{x}+e_{t0}]\\ \left(\hat{C}^{\prime\prime}_{pk0}-\frac{d_{l}t_{\alpha/4\mu-1}}{3\sqrt{n}} \right)\times\sqrt{\frac{\chi_{2/4\mu-1}^{2}}{n-1}},\ \overline{x}+e_{t0}<T\\ \left(\hat{C}^{\prime\prime}_{pk0}-\frac{d_{l}t_{\alpha/4\mu-1}}{3\sqrt{n}} \right)\times\sqrt{\frac{\chi_{2/4\mu-1}^{2}}{n-1}},\ \overline{x}-e_{t0}>T\\ \end{array} \right., \tag{30}\]  
\[UC^{\prime\prime}_{pk}=\left\{ \begin{array}{l}\hat{C}^{\prime\prime}_{pk0} \times\sqrt{\frac{\chi_{1-\alpha/4\mu-1}^{2}}{n-1}},\ T\in[\overline{x}-e_{t0 },\ \overline{x}+e_{t0}]\\ \left(\hat{C}^{\prime\prime}_{pk0}+\frac{d_{l}t_{\alpha/4\mu-1}}{3\sqrt{n}} \right)\times\sqrt{\frac{\chi_{1-\alpha/4\mu-1}^{2}}{n-1}},\ \overline{x}+e_{t0}<T\\ \left(\hat{C}^{\prime\prime}_{pk0}+\frac{d_{\alpha/4\mu-1}}{3\sqrt{n}} \right)\times\sqrt{\frac{\chi_{1-\alpha/4\mu-1}^{2}}{n-1}},\ \overline{x}-e_{t0}>T\\ \end{array} \right., \tag{31}\]  
where  
\[\hat{C}^{\prime\prime}_{pk0}=\left\{ \begin{array}{l}\frac{d}{3s},\ T\in[ \overline{x}-e_{t0},\ \overline{x}+e_{t0}]\\ \frac{d+d_{l}\times(\overline{x}-T)}{3s},\ \overline{x}+e_{t0}<T\\ \frac{d-d_{\mu}\times(\overline{x}-T)}{3s},\ \overline{x}-e_{t0}>T\\ \end{array} \right.. \tag{32}\]  
[PERSON] et al. [21] defined the following levels for process capability requirements:  
1. Process capability is considered \"inadequate\" when \(C^{\prime\prime}_{pk}<1.00\). 2. Process capability is considered \"capable\" when \(1.00\leq C^{\prime\prime}_{pk}<1.33\)
3. Process capability is considered \"satisfactory\" when \(1.33\leq C^{\prime\prime}_{pk}<1.5\)
4. Process capability is considered \"excellent\" when \(1.50\leq C^{\prime\prime}_{pk}<2.00\)
5. Process capability is considered \"superb\" when \(2.00\leq C^{\prime\prime}_{pk}\)  
To determine whether asymmetric tolerances index \(C^{\prime\prime}_{pk}\) is equal to \(c\), we adopt null hypothesis \(H_{0}\):\(C^{\prime\prime}_{pk}=C\) against alternative hypothesis \(H_{1}\):\(C^{\prime\prime}_{pk}\
eq C\). Based on Equations (26) and (27) and with significance level \(\alpha\), we construct the following three statistical testing rules:  
1. If \(LC^{\prime\prime}_{pk}>C\), then reject \(H_{0}\) and conclude that \(C^{\prime\prime}_{pk}>C\) (i.e., consider lowering costs). 2. If \(UC^{\prime\prime}_{pk}<C\), then reject \(H_{0}\) and conclude that \(C^{\prime\prime}_{pk}<C\) (i.e., improve the process). 3. If \(LC^{\prime\prime}_{pk}<C<UC^{\prime\prime}_{pk}\), then accept \(H_{0}\) and presume \(C^{\prime\prime}_{pk}=C\) (i.e., maintain status quo). Next, to help the industry successfully apply the proposed assessment model, we established the following procedure:  
Step 1: Based on the capability level required by customers, determine the C value, significance level \(\alpha\), and sample size \(n\).