## 3 Construct the Measurement Coordinate Point  
This article established the PCAC-AT analysis chart in Section 2 and then will establish the rules based on the analysis chart of Section 2, and then construct the evaluation coordinate points of each quality characteristic according to the principle of statistical inference and evaluation rules. Let \(X_{h,1},\ldots,X_{h,n}\) be a random sample derived from quality characteristic h with sample size \(n\), where \(h=1\), \(2\), \(\ldots\), \(k\). Then the estimators of \(\mu_{h}\) is \(\overline{X}_{h}=\frac{1}{n}\sum_{j=1}^{n}X_{h,j}\) and the estimators of \(\sigma_{h}\) is \(S_{h}=\sqrt{\frac{1}{n}\sum_{j=1}^{n}\left(X_{h,j}-\overline{X}_{h}\right)^{2}}\). Therefore, the estimators of index \(\delta_{h}^{\prime}\) is \(\hat{\delta}_{h}^{\prime}=\left(\overline{X}_{h}-T_{h}\right)/d_{h}^{*}\) and the estimators of index \(\gamma_{h}^{\prime}\) is \(\hat{\gamma}_{h}^{\prime}=S_{h}/d_{h}^{*}\). With an assumption of normality, \(T_{h}=\sqrt{n}\left(\hat{\delta}_{h}^{\prime}-\delta_{h}^{\prime}\right)/ \hat{\gamma}_{h}\) is distributed as a t-distribution with n\(-\)1 degree of freedom, denoted by \(h_{n-1}\), and \(K_{h}=n\hat{\gamma}_{h}^{2}/\gamma_{h}^{2}\) is distributed as a chi-square distribution with n\(-\)1 degree of freedom, denoted by \(\chi_{n-1}^{n}\). Since the probability that \(T_{h}\) between \(-t_{1-\alpha/4n-1}\) and \(t_{1-\alpha/4n-1}\) is \(1-\alpha/2\), where \(t_{\alpha/4n-1}\) is the upper \(\alpha/4\) quintile of \(t_{n-1}\). Therefore, the \(1-\alpha/2\) lower confidence limit of \(\delta_{h}^{\prime}\) is \(\hat{\delta}_{h}^{\prime}+\epsilon_{h}\), where \(e_{h}=t_{\alpha/4n-1}\hat{\gamma}_{h}/\sqrt{n}\). Similarly, since probability that \(K_{h}\) between \(\chi_{n/4n-1}^{2}\) and \(\chi_{1-\alpha/4n-1}^{2}\) is \(1-\alpha/2\), where \(\chi_{n/4n-1}^{2}\) is the lower \(\alpha/4\) quintile of \(\chi_{n-1}^{2}\), and \(\chi_{1-\alpha/4n-1}^{2}\) is the lower \(1-\alpha/4\) quintile of \(\chi_{n-1}^{2}\). Therefore, the \(1-\alpha/2\) lower confidence limit of \(\gamma_{h}^{\prime}\) is \(\gamma_{Lh}^{\prime}\) and the \(1-\alpha/2\) upper confidence limit of \(\gamma_{h}^{\prime}\) is \(\gamma_{Uh}^{\prime}\), where \(\gamma_{Uh}^{\prime}=\sqrt{n/\chi_{1-\alpha/4n-1}^{2}}\hat{\gamma}^{\prime}\) and \(\gamma_{Uh}^{\prime}=\sqrt{n/\chi_{2/4/4n-1}^{2}}\hat{\gamma}^{\prime}_{h}\). Obviously, when \(\hat{\delta}_{h}^{\prime}-e_{h}\geq 0\), it can be inferred that \(\delta_{h}^{\prime}\geq 0\) and \(\mu_{h}\geq T_{h}\). Likewise, when \(\delta_{h}^{\prime}+e_{h}<0\), it can be inferred that \(\delta_{h}^{\prime}<0\) and \(\mu_{h}<T_{h}\). Thus,  
\[\hat{\delta}_{h}=\left\{ \begin{array}{l}\delta_{h}^{\prime}d_{h1},\delta_{h}^{ \prime}-e_{h}\geq 0\\ \delta_{h}^{\prime}d_{h2},\delta_{h}^{\prime}+e_{h}<0\end{array} \right.. \tag{7}\]  
In order to derive the \(100(1-\alpha)\%\) confidence region of \(\left(\delta_{h}^{\prime},\gamma_{h}^{\prime}\right)\), this paper defines events \(E_{Sh}=\left\{\delta_{Lh}^{\prime}\leq\delta_{h}^{\prime}\leq\delta_{Uh}^{ \prime}\right\}\) and \(E_{\gamma h}=\left\{\gamma_{LL}^{\prime}\leq\gamma_{h}^{\prime}\leq\gamma_{Uh}^{ \prime}\right\}\). Obviously, \(p(E_{\delta h})=p\big{(}E_{\gamma h}\big{)}=1-\alpha/2\) and \(p\big{(}E_{Sh}^{C}\big{)}=p\Big{(}E_{\gamma h}^{C}\big{)}=\alpha/2\). Based on Boole's inequality and DeMorgan's rules, we have \(p\big{(}E_{Sh}\cap E_{\gamma h}\big{)}\geq 1-p\big{(}E_{Sh}^{C}\big{)}-p\Big{(}E_{ \gamma h}^{C}\big{)}\)\(=\alpha/2\). Then, \(CR=\big{\{}\left(\delta_{h},\gamma_{h}\right)|\delta^{t}_{Lh}\times d_{h1}\leq \delta_{h}\leq\delta^{t}_{ULh}\times d_{h1},\gamma^{\prime}_{Lh}\times d_{h1} \leq\gamma_{h}\leq\gamma^{\prime}_{UL}\times d_{h1}\big{\}}\), where \(\hat{\gamma}_{h}=\hat{\gamma}^{\prime}_{h}d_{h1}\), \(\delta_{h}=\delta^{\prime}_{h}d_{h1}\). is the \(100(1-\alpha)\%\) confidence region of \((\delta_{h},\gamma_{h})\) for \(\delta^{\prime}_{Lh}\geq 0\).