## 3 Evaluation of Process Precision and Accuracy
Inferring from their relationships, we have the following equation:  
\[p\Bigl{\{}-Z_{\alpha^{\prime}/2}\leq Z_{mn}\leq Z_{\alpha^{\prime}/2},\chi_{ \alpha^{\prime}/2;N-m}^{2}\leq K_{mn}\leq\chi_{1-(\alpha^{\prime}/2);N-m}^{2} \Bigr{\}}=1-\alpha \tag{34}\]  
Equivalently,  
\[p\Biggl{\{}\delta-Z_{\alpha^{\prime}/2}\times\biggl{(}\frac{\gamma}{\sqrt{N} }\biggr{)}\leq\delta\leq\hat{\delta}+Z_{\alpha^{\prime}/2}\times\biggl{(} \frac{\gamma}{\sqrt{N}}\biggr{)},\sqrt{\frac{N-m}{\chi_{1-(\alpha^{\prime}/2); N-m}^{2}}}\hat{\gamma}\leq\gamma\leq\sqrt{\frac{N-m}{\chi_{\alpha^{\prime}/2; N-m}^{2}}}\hat{\gamma}\Biggr{\}}=1-\alpha \tag{35}\]  
According to Equations (20) and (30), the observation value of \(\hat{\delta}\) is \(\overline{\delta}\) and the observation value of \(\hat{\gamma}\) is \(\overline{\gamma}\), where \(\overline{\delta}=\sum_{j=1}^{m}\overline{y}_{j}\) and \(\overline{\gamma}=b_{11}^{-1}\overline{s}\). Obviously, the 100 \((1-\alpha)\%\) confidence region of \((\delta,\gamma)\) is similar to a trapezoid, wide at the top and narrow at the bottom. Therefore, this study defines \(CR_{\delta}=[\delta_{L},\delta_{R}]\) and \(CR_{\gamma}=[\gamma_{L},\gamma_{R}]\), where  
\[\delta_{L}=\overline{\delta}-Z_{0.5-\sqrt{1-\alpha}/2}\times\sqrt{\frac{N-m}{ N\chi_{0.5-\sqrt{1-\alpha}/2;N-m}^{2}}}\overline{\gamma} \tag{36}\]  
\[\delta_{R}=\overline{\delta}+Z_{0.5-\sqrt{1-\alpha}/2}\times\sqrt{\frac{N-m}{ N\chi_{0.5-\sqrt{1-\alpha}/2;N-m}^{2}}}\overline{\gamma} \tag{37}\]  
\[H_{1}:\delta\
otin A_{\delta}\;\sigma\;\gamma\
otin A_{\gamma} \tag{29}\]0  
\[H_{1}:\delta\
otin A_{\delta}\;\sigma\;\gamma\
otin A_{\gamma} \tag{29}\]1  
Then, the \(100\)\((1-\alpha)\)% lower confidence limit \(\gamma_{L}\) is adopted to establish the following testing rules:  
1. If \(\gamma_{L}>1/k\), then \(H_{0}\) is rejected. It is concluded that precision index \(\gamma\) is bigger than \(1/k\), indicating that the process variation is so huge that the process must be improved. 2. If \(\gamma_{L}\leq 1/k\), then \(H_{0}\) is not rejected. It is concluded that precision index \(\gamma\) is smaller than or equal to \(1/k\), showing that the process variation does not need to reduce. When the accuracy index is bigger than \(1.5/k\), then it is learned that the process is overly shifted to the right of tolerance, so it must be improved. In contrast, as the value of accuracy index is smaller than \(-1.5/k\), it is known that the process is overly shifted to the left of tolerance, so it must be bettered as well. Therefore, for the statistical test of accuracy index \(\delta\), the null hypothesis (\(H_{0}^{\prime\prime}\)) and the alternative hypothesis (\(H^{\prime\prime}\)) are depicted as follows:  
\[H_{1}:\delta\
otin A_{\delta}\;\sigma\;\gamma\
otin A_{\gamma} \tag{29}\]2  
\[H_{1}:\delta\
otin A_{\delta}\;\sigma\;\gamma\
otin A_{\gamma} \tag{29}\]3  
Next, the \(100\)\((1-\alpha)\)% lower confidence limit \(\delta_{L}\) and upper confidence limit \(\delta_{R}\) are applied to the testing rules as follows:  
1. If \([\delta_{L},\delta_{R}]\cap[-1.5/k,1.5/k]\
eq\phi\), then do not reject \(H_{0}\), and the process accuracy does not need to be improved. 2. If \(\delta_{L}>1.5/k\), indicating that the right deviation of the process exceeds \(1.5\) sigma, then it is necessary to make some improvements to help the process mean move toward the target value. Also, the process deviation must be controlled and fall within \(1.5\) sigma or less. 3. If \(\delta_{R}<-1.5/k\), meaning that the left deviation of the process exceeds \(1.5\) sigma, then it is necessary to make some adjustments to help the process mean move toward the target value. Also, the process deviation must be controlled and fall within \(1.5\) sigma or less. **5. An Empirical Example**  
As mentioned above, Process Capability Indices (PCIs) are tools for process quality evaluations that are widely applied in the industry, including various machine tools and machining industries, semiconductor processes, packaging processes, etc.