# Fuzzy Quality Evaluation Model Constructed by Process Quality Index
## 4 Developing a Fuzzy Hypothesis Testing Method  
Before the fuzzy hypothesis testing method is developed, the process of statistical hypothesis testing is examined. According to [PERSON] et al. [20], we adopt the following hypotheses when judging if the quality level meets the required value \(k\):  
null hypothesis \(H_{0}\):\(P_{QI}\)\(\geq\)\(k\)  
versus  
alternative hypothesis \(H_{1}\):\(P_{QI}\)\(<\)\(k\). Next, the critical value is determined by  
\[ \begin{array}{l}1-\alpha~{}=~{}p\Big{\{}P_{QI}^{*}<C_{0}\Big{|}P_{QI}\geq k \Big{\}}\\ ~{}=~{}p\Big{\{}\sqrt{NP_{QI0}^{*}}<\sqrt{N}C_{0}\Big{|}P_{QI}\geq k\Big{\}} \\ ~{}=~{}p\Big{\{}t_{N-m}\Big{(}\Delta~{}=~{}\sqrt{N}k\Big{)}<\sqrt{N}C_{0}\Big{\}},\end{array}  \tag{23}\]  
where \(t_{N-m}\Big{(}\Delta~{}=~{}\sqrt{N}k\Big{)}\) is the non-central t distribution with \(N-m\) degrees of freedom and non-centrality parameter \(\Delta~{}=~{}\sqrt{N}k\). Thus, the critical value \(C_{0}\) is denoted as follows:  
\[C_{0}=~{}\frac{t_{\alpha;N-m}\Big{(}\Delta~{}=~{}\sqrt{N}k\Big{)}}{\sqrt{N}}, \tag{24}\]  
where \(t_{\alpha;N-m}\Big{(}\Delta~{}=~{}\sqrt{N}k\Big{)}\) is the lower \(\alpha\)th quantile of \(t_{N-m}\Big{(}\Delta~{}=~{}\sqrt{N}k\Big{)}\). In applying the critical value \(C_{0}\) to statistical hypothesis testing, the rules are illustrated below:  
(1) When \(P_{QI0}^{*}\)\(<\)\(C_{0}\), then reject \(H_{0}\) and conclude that \(P_{QI}\)\(<\)\(k\). (2) When \(P_{QI0}^{*}\)\(\geq\)\(C_{0}\), then do not reject \(H_{0}\) and conclude that \(P_{QI}\)\(\geq\)\(k\). On the basis of Equation (17), the \(\alpha\)-cuts of the triangular fuzzy number \(\widetilde{C}_{0}\) are expressed as follows:  
\[\widetilde{C}_{0}[\alpha]~{}=~{}\left\{ \begin{array}{l}[C_{0}(1),C_{0}( \alpha)]_{,}~{}\text{for}~{}0.01\leq\alpha\leq 1\\ [C_{0}(1),C_{0}(0.01)]_{,}~{}\text{for}~{}0\leq\alpha\leq 0.01\end{array} \right., \tag{25}\]where  
\[C_{0}(1)=\ \left(C_{0}-\frac{t_{0,5;N-m}}{\sqrt{mn}}\right)\sqrt{\frac{\chi_{0,5;N-m}^{2}}{\chi_{0,5;N-m}^{2}}}\ =\ C_{0} \tag{26}\]  
and  
\[C_{0}(\alpha)=\ \left(C_{0}+\frac{t_{\alpha/2;N-m}}{\sqrt{mn}}\right)\sqrt{\frac{ \chi_{1-\alpha/2;N-m}^{2}}{\chi_{0,5;N-m}^{2}}} \tag{27}\]  
Based on Equations (26) and (27), the half-triangle-shaped fuzzy number of \(\widetilde{C}_{0}\) is \(\Delta\widetilde{C}_{0}\ =\ (C_{M},C_{R})\), where  
\[C_{M}\ =\ C_{0} \tag{28}\]  
and  
\[C_{R}\ =\ \left(C_{0}+\frac{t_{0,005;N-m}}{\sqrt{mn}}\right)\sqrt{\frac{\chi_{0,995;N-m}^{2}}{\chi_{0,5;N-m}^{2}}}. \tag{29}\]  
Then, the fuzzy membership function of \(\widetilde{C}_{0}\) is defined below:  
\[\eta(x)\ =\ \left\{ \begin{array}{ll}0\ \ \ if\ \ x<C_{M}\\ 1\ \ if\ \ x=C_{M}\\ \alpha\ \ if\ \ C_{M}<x\leq C_{R}\\ 0\ \ if\ \ C_{R}<x\end{array} \right., \tag{30}\]  
where \(\alpha\) is determined by  
\[\left(C_{0}+\frac{t_{\alpha/2;N-m}}{\sqrt{mn}}\right)\sqrt{\frac{\chi_{1- \alpha/2;N-m}^{2}}{\chi_{0,5;N-m}^{2}}}\ =\ x \tag{31}\]  
Obviously, the critical value \(C_{0}\) is the subject of the evaluation, and the estimated value of the index varies. Therefore, it is assumed that set \(A_{T}\) be the area in the graph of \(\eta(x)\) and set \(A_{R}\) is the area in the graph of \(\eta(x)\) but to the right of the vertical line \(x=\ P_{QI0}^{*}\). Figure 2 displays a diagram of membership functions \(\eta(x)\) and \(\eta_{0}(x)\). Thus,  
\[A_{T}\ =\ \{\ (x,\alpha)|C_{0}(1)\leq x\leq C_{0}(\alpha),0\leq\alpha\leq 1\} \tag{32}\]  
Figure 2: Diagram of membership functions \(\eta(x)\) and \(\eta_{0}(x)\). \[A_{R}\ =\ \{\ (x,\alpha)|Q^{*}_{P10}\leq x\leq C_{0}(\alpha),0\leq\alpha\leq a\}, \tag{33}\]  
where \(x(a)\ =\ P^{*}_{Q10}\). According to [PERSON] [26], the area of \(A_{R}\) is adopted as the numerator and the area of \(A_{T}\) as the denominator. Besides, the fuzzy test is performed based on the ratio of \(A_{R}/A_{T}\). Some scholars think that the calculation of this ratio is relatively difficult and therefore can be replaced by the bottom-line length ratio [27; 28]. First, let \(d_{T}\ =\ C_{R}-C_{M}\) be the bottom length of \(\eta(x)\).