## 2 Materials and Methods
### Confidence Interval Based Fuzzy Numbers  
As mentioned previously, the proposed scheme can derive the SSQI for the IC packaging molding process, which further enables development of a fuzzy model. An estimator of index accuracy \(\delta_{h}\) and an estimator of index precision \(\gamma_{h}\) are used to derive confidence interval based fuzzy numbers for the SSQI as well as the membership function for development of the fuzzy evaluation model. We assume that the random variables \(Y_{h}\sim N\Big{(}\delta_{h},\gamma_{h}^{2}\Big{)}\) represent the process distribution of quality characteristic \(h\), where \(Y_{h1}\),..., \(Y_{hi}\),..., \(Y_{lm}\) is a set of sample data related to quality characteristic \(h\), and the sample size is \(n\). The estimator of index \(Q_{PUh}\) for quality characteristic \(h\) can be derived using these sample data, as follows:  
\[Q_{PUh}^{*}=\frac{1-\delta_{h}^{*}}{\gamma_{h}^{*}}+1.5, \tag{11}\]  
where \(\delta_{h}^{*}=n^{-1}\times\sum_{i=1}^{n}Y_{hi}\) and \(\gamma_{h}^{*}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(Y_{hi}-\delta_{h}^{*} \right)^{2}}\). Similarly, if we let  
\[Z=\sqrt{n}\Bigg{[}\big{(}Q_{PUh}-1.5\big{)}-\big{(}Q_{PUh}^{*}-1.5\big{)} \times\left(\frac{\gamma_{h}^{*}}{\gamma_{h}}\right)\Bigg{]}=\frac{\sqrt{n} \Big{(}\delta_{h}^{*}-\delta_{h}\Big{)}}{\gamma_{h}}, \tag{12}\]  
then \(Z\) follows standard normal distribution; i.e., \(Z\sim N(0,1)\). Therefore,  
\[p\Bigg{\{}\sqrt{n}\Bigg{[}\big{(}Q_{PUh}-1.5\big{)}-\big{(}Q_{PUh}^{*}-1.5 \big{)}\times\left(\frac{\gamma_{h}^{*}}{\gamma_{h}}\right)\Bigg{]}\leq Z_{ \alpha/2}\Bigg{\}}=1-\frac{\alpha}{2}.\]  
Equivalently,  
\[P\Bigg{\{}Q_{PUh}\leq\big{(}Q_{PUh}^{*}-1.5\big{)}\times\left(\frac{\gamma_{ h}^{*}}{\gamma_{h}}\right)+\frac{Z_{\alpha/2}}{\sqrt{n}}+1.5\Bigg{\}}=1-\frac{ \alpha}{2}, \tag{13}\]  
where \(Z_{\alpha/2}\) is the upper \(\alpha/2\) quantile of standard normal distribution. Conversely, if we let  
\[K=\frac{(n-1){\gamma_{h}^{*}}^{2}}{\gamma_{h}^{2}}=\frac{\sum_{i=1}^{n}\left( Y_{hi}-\delta_{h}^{*}\right)^{2}}{\gamma_{h}^{2}}, \tag{14}\]  
then \(K\) presents a chi-square distribution with \(n-1\) degrees of freedom (i.e., \(\chi_{n-1}^{2}\)). Thus, we can further derive the following:  
\[p\Bigg{(}\frac{(n-1){\gamma_{h}^{*}}^{2}}{\gamma_{h}^{2}}\leq\chi_{1-\alpha/2,n-1}^{2}\Bigg{)}=1-\frac{\alpha}{2}.\]  
Equivalently,  
\[p\Bigg{(}\frac{\gamma_{h}^{*}}{\gamma_{h}}\leq\sqrt{\frac{\chi_{1-\alpha/2,n- 1}^{2}}{n-1}}\Bigg{)}=1-\frac{\alpha}{2}, \tag{15}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Quality Level** & \(\omega\) & \(\omega^{\prime}\) \\ \hline \(6\sigma\) & 6.00 & 6.23 \\ \(5\sigma\) & 5.00 & 5.28 \\ \(4\sigma\) & 4.00 & 4.37 \\ \(3\sigma\) & 3.00 & 3.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of quality levels for \(\omega\) and \(\omega^{\prime}\). where \(\chi^{2}_{1-\alpha/2,n-1}\) is the lower \(\alpha/2\) quantile of the chi-square distribution with \(n-1\) degrees of freedom. In fact, index \(Q_{PUh}\) is a function of \((\delta_{h},\gamma_{h})\). To derive the \((1-\alpha)\times 100\%\) upper confidence limits on index \(Q_{PUh}\), this paper defines events \(A\) and \(B\) of \((\delta_{h},\gamma_{h})\) as follows:  
\[A=\bigg{\{}(\delta_{h},\gamma_{h})\bigg{|}Q_{PUh}\leq\Big{(}Q_{PUh}-1.5\Big{)} \times\bigg{(}\frac{\gamma_{h}^{*}}{\gamma_{h}}\Big{)}+\frac{Z_{\alpha/2}}{ \sqrt{n}}+1.5\bigg{\}};B=\Bigg{\{}(\delta_{h},\gamma_{h})\bigg{|}\frac{\gamma_ {h}^{*}}{\gamma_{h}}\leq\sqrt{\frac{\chi^{2}_{1-\alpha/2,n-1}}{n-1}}\Bigg{\}}.\]  
Clearly, \(p(A)=p(B)=1-\alpha/2\) and \(p\Big{(}A^{C}\Big{)}=p\Big{(}B^{C}\Big{)}=\alpha/2\). According to [PERSON]'s theorem, we have \(\Big{(}A^{C}\cup B^{C}\Big{)}^{C}=A\cap B\) and \(p(A\cap B)=1-p\Big{(}A^{C}\cup B^{C}\Big{)}\geq 1-p\Big{(}A^{C}\Big{)}-p \Big{(}B^{C}\Big{)}=1-\alpha\).