## 2 Materials and Methods
### Fuzzy Evaluation Model  
Before developing the fuzzy hypothesis testing method, we examined the process of statistical hypothesis testing. In determining whether the index \(Q_{PUH}\) was at least equal to \(\omega^{\prime}\), we adopted the null hypothesis as \(H_{0}:Q_{PUH0}\geq\omega^{\prime}\) and the alternative hypothesis as \(H_{1}:Q_{PUH0}<\omega^{\prime}\). For the upper confidence limit \(UQ_{PUH0}\), the rules of statistical hypothesis testing were as follows:  
* If \(UQ_{PUH0}\)\(\geq\)\(\omega^{\prime}\), then do not reject \(H_{0}\), and conclude that \(Q_{PUH}\geq\omega^{\prime}\). * If \(UQ_{PUH0}\)\(<\)\(\omega^{\prime}\), then reject \(H_{0}\), and conclude that \(Q_{PUH0}<\omega^{\prime}\). In fact, \(UQ_{PUH0}\) is a decreasing function of sample size \(n\), which can be rewritten as \(UQ_{PUH0}(n)\). Let \(n^{\prime}<n^{\prime}\) such that \(UQ_{PUH0}(n^{\prime})\)\(\geq\)\(\omega^{\prime}\) and \(UQ_{PUH0}(n^{\prime})\)\(<\)\(\omega^{\prime}\), then  
* \(UQ_{PUH0}(n^{\prime})\)\(\geq\)\(\omega^{\prime}\), do not reject \(H_{0}\), and conclude that \(UQ_{PUH0}\)\(\geq\)\(\omega^{\prime}\). * \(UQ_{PUH0}(n^{\prime})\)\(<\)\(\omega^{\prime}\), reject \(H_{0}\), and conclude that \(UQ_{PUH0}<\omega^{\prime}\). It is clear that sample size \(n\) influences the statistical inference results. Thus, this paper will be in accordance with [21] and [22] to propose a fuzzy hypothesis testing method. For the sake of convenience, we referred to [22] and replaced the area of membership function with the bottom line of membership function. Based on this concept, we developed a novel approach to fuzzy testing involving \(H_{0}:Q_{PUH}\geq\omega^{\prime}\) and \(H_{1}:Q_{PUH}<\omega^{\prime}\). Figure 1 presents a diagram corresponding to the membership function \(\eta_{h}(x)\) and vertical line \(x=\omega^{\prime}\). Figure 1: Diagram corresponding to \(\eta_{h}(x)\) and vertical line \(x=\omega^{\prime}\). We can compute the total area \(a_{Th}\) of \(\eta_{h}(x)\) as follows:  
\[a_{Th}=\int\limits_{Q_{Mh}}^{Q_{Rh}}\eta_{h}(x)\ d_{x}. \tag{23}\]  
Similarly, we can compute the slashed area \(a_{Rh}\) of \(\eta_{h}(x)\) as follows:  
\[a_{Rh}=\int\limits_{\alpha^{\prime}}^{Q_{Rh}}\eta(x)\ d_{x}. \tag{24}\]  
According to [21], we can use \(a_{Rh}\) as the numerator and \(2a_{Th}\) as the denominator and then perform fuzzing testing using \(a_{Rh}/2a_{Th}\). Supposing that \(d_{Rh}=Q_{Rh}-\omega^{\prime}\), \(d_{Rh}\) is the base of slashed area \(\eta_{h}(x)\). Based on Equation (20), \(d_{Rh}\) can be shown as follows:  
\[d_{Rh}=\left(Q_{PULh0}^{*}-1.5\right)\times\sqrt{\frac{x_{0.975,n-1}^{2}}{n-1 }}+\frac{Z_{0.025}}{\sqrt{n}}-(\omega^{\prime}-1.5). \tag{25}\]  
Similarly, the base of \(\eta_{h}(x)\) can be expressed as \(d_{Th}=Q_{Rh}-Q_{Mh}\). Based on Equations (19) and (20), \(d_{Th}\) can be shown as follows:  
\[d_{Th}=\left(Q_{PULh0}^{*}-1.5\right)\times\left(\sqrt{\frac{x_{0.975,n-1}^{2} }{n-1}}-\sqrt{\frac{x_{0.5,n-1}^{2}}{n-1}}\right)+\frac{Z_{0.025}}{\sqrt{n}}. \tag{26}\]  
For the sake of convenience, we referred to [22] and replaced \(a_{Rh}/2a_{Th}\) with \(d_{Rh}/2d_{Th}\) for fuzzy testing; \(d_{Rh}/2d_{Th}\) is defined as follows:  
\[d_{Rh}/2d_{Th}=\left\{ \begin{array}{ll}0&if\ \omega^{\prime}\geq Q_{Rh}\\ \frac{Q_{Rh}-\omega^{\prime}}{2(Q_{Rh}-Q_{Mh})}&if\ Q_{Mh}<\omega^{\prime}<Q_ {Rh}\\ 1/2&if\ \omega^{\prime}\leq Q_{Mh}\end{array} \right.. \tag{27}\]  
If we let \(0\ <\phi_{1}<\phi_{2}<0.5\), in accordance with [21], we can consider employing two numbers as follows:  
1. If \(d_{Rh}/2d_{Th}\leq\phi_{1}\), then we reject \(H_{0}\), and we conclude that \(Q_{PULh}<\omega^{\prime}\). 2. If \(\phi_{1}<d_{Rh}/2d_{Th}<\phi_{2}\), then we make no decision regarding whether to reject/not reject. 3. If \(\phi_{2}\leq d_{Rh}/2d_{Th}\leq 0.5\), then we do not reject \(H_{0}\), and we conclude that \(Q_{PULh}\geq\omega^{\prime}\). Figure 2 illustrates the procedures of the proposed fuzzy evaluation model for the IC packaging molding process.