## 3 Evaluation of Process Precision and Accuracy  
As mentioned earlier, this study used the \(\delta-\gamma\) control charts derived in Section 2 to monitor the process precision and accuracy. As the process was statistically controlled, this study used the statistical testing method as well as applied the control chart data in statistical process control, in order to propose a model to evaluate process precision and accuracy. Plenty of studies have suggested that as the process quality attains the \(k\)-sigma quality level, the required conditions are (1) \(|\delta|\leq 1.5/k\) and (2) \(\gamma\leq 1/k\)[20]. That is, when the accuracy index is bigger than \(1.5/k\) (\(\delta>1.5/k\)), it is learned that the process is excessively shifted to the right of tolerance, so the process must be improved. On the other hand, when the accuracy index is smaller than \(-1.5/k\) (\(\delta<-1.5/k\)), it indicates that the process is excessively shifted to the left of tolerance, so the process must make some improvements. Likewise, when the precision index is bigger than \(1/k\) (\(\gamma>1/k\)), it shows that the process variation is too big and must be modified. Therefore, if the process quality level is required to reach \(k\)-sigma, then the null hypothesis (\(H_{0}\)) and the alternative hypothesis (\(H_{1}\)) for statistical tests are expressed as follows:  
\[H_{0}:\delta\in A_{\delta}\text{ {and} }\gamma\in A_{\gamma} \tag{28}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Subsample Size \(n\)** & \(b_{n}\) & \(A_{n}\) & \(B_{n}\) & \(B^{\prime}_{n}\) \\ \hline
6 & 0.869 & 1.410 & 1.970 & 0.030 \\
7 & 0.888 & 1.277 & 1.882 & 0.118 \\
8 & 0.903 & 1.175 & 1.815 & 0.185 \\
9 & 0.914 & 1.094 & 1.761 & 0.239 \\
10 & 0.923 & 1.028 & 1.716 & 0.284 \\
11 & 0.930 & 0.973 & 1.679 & 0.321 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of \(b_{n}\), \(A_{n}\), \(B_{n}\), and \(B^{\prime}_{n}\) for subsample size \(n=6(1)11\). \[H_{1}:\delta\
otin A_{\delta}\;\sigma\;\gamma\
otin A_{\gamma} \tag{29}\]  
where \(A_{\delta}=\{-1.5/k\leq\delta\leq 1.5/k\}\) and \(A_{\gamma}=\{\gamma\leq 1/k\}\). Then, the 100 \((1-\alpha)\%\) confidence region of \((\delta,\gamma)\) is received to establish testing rules. In statistical process control, the unbiased estimators of \(\delta\) and \(\gamma^{2}\), respectively, are expressed as follows:  
\[\hat{\delta}=\frac{1}{m}\sum_{j=1}^{m}\overline{\Upsilon}_{j}=\frac{1}{mn} \sum_{j=1}^{m}\sum_{i=1}^{n}\gamma_{j,i} \tag{30}\]  
\[\hat{\gamma}^{2}=\frac{1}{m(n-1)}\sum_{j=1}^{m}\sum_{i=1}^{n}\bigl{(}\gamma_{ j,i}-\overline{\Upsilon}_{j}\bigr{)}^{2} \tag{31}\]  
Furthermore, let random variables be \(Z_{mn}=\sqrt{mn}\bigl{(}\delta-\delta\bigr{)}/\gamma\) and \(K_{mn}=m(n-1)\hat{\gamma}^{2}/\gamma^{2}\). Given the assumption of normality, \(Z_{mn}\) and \(K_{mn}\) are, respectively, distributed as \(N(0,1)\) and \(\chi_{N-m}^{2}\). Therefore,  
\[p\{-Z_{\alpha^{\prime}/2}\leq Z_{mn}\leq Z_{\alpha^{\prime}/2}\}=\sqrt{1-\alpha} \tag{32}\]  
\[p\Bigl{\{}\chi_{\alpha^{\prime}/2;N-m}^{2}\leq K_{mn}\leq\chi_{1-(\alpha^{ \prime}/2);N-m}^{2}\Bigr{\}}=\sqrt{1-\alpha} \tag{33}\]  
where \(N=mn\), \(Z_{\alpha^{\prime}/2}\) is the upper \(\alpha^{\prime}/2\) quantile for a standard normal distribution, \(\chi_{\alpha^{\prime}/2;N-m}^{2}\) is the upper \(\alpha^{\prime}/2\) quantile of \(\chi_{N-m}^{2}\), and \(\alpha^{\prime}=1-\sqrt{1-\alpha}\). \(\hat{\delta}\) and \(\hat{\gamma}^{2}\) are mutually independent, and so are \(Z_{mn}\) and \(K_{mn}\).