## Appendix A Derivation of Sample Size Equation for Redesign of Mechanical Systems through ALT  
To acquire the number of mission cycles of a parametric ALT from the target BX lifetime on the test scheme, the sample size formulation integrated with acceleration factors needs to be determined. Currently, numerous methods have been suggested to settle matters on sample size. The Weibayes model, utilizing Weibull examination, is a popularly accepted procedure of inspecting reliability data. However, it cannot be straightforwardly utilized due to the complication of the mathematical equation. The cases as failures (\(r\geq 1\)) and no failures (\(r=0\)) need to be separate. As a consequence, it is practical to obtain a probable sample size formulation that could provide the mission cycle after right presumptions. The Weibull distribution is popularly utilized because it can be simply described as a shape parameter and characteristic life. That is, if the product pursues Weibull distribution, the accumulated failure rate, \(F(t)\), in Equation (2) is stated as  
\[F(t)=1-e^{-(\frac{t}{\eta})^{\beta}} \tag{1}\]  
where \(t\) is time, \(\eta\) is characteristic life, and \(\beta\) is shape parameter. When \(t=L_{B}\) in Equation (A1), the correlation between BX life, \(L_{B}\), and characteristic life, \(\eta\), may be stated as  
\[L_{B}^{\beta}=\left(ln\frac{1}{1-x}\right)\cdot\eta^{\beta}\] (A2)  
where \(x=0.01F(t)\). Product failures in the bathtub curve might be categorized into three sections: infant mortality (\(\beta<1\)), random failure (\(\beta=1\)), and wear-out failure ((\(\beta>1\)) that may be defined as shape parameter in the Weibull distribution. For approximating lifetime, the Weibayes method is explained as Weibull analysis with an assigned shape parameter that can be presumed from prior learning or actual testing. For a Weibayes analysis, the characteristic life can be obtained from utilizing the maximum likelihood estimate (\(\mathit{MLE}\)) as follows:  
\[\eta_{\mathit{MLE}}^{\beta}=\frac{\sum_{i=1}^{n}t_{i}^{\beta}}{r}\] (A3)  
where \(\eta_{\mathit{MLE}}\) is the \(\mathit{MLE}\) of the characteristic life, \(n\) is the whole number of samples, \(t_{i}\) is the full test length for each sample, and \(r\) is the number of failures. When the failure number is \(r\geq 1\) and confidence level is 100(1-\(\alpha\)), the characteristic life, \(\eta_{\alpha}\), could be approximated as the following middle term and altered into the last term utilizing Equation (A3):  
\[\eta_{\alpha}^{\beta}=\frac{2r}{\chi_{\alpha}^{2}(2r+2)}\cdot\eta_{\mathit{ MLE}}^{\beta}=\frac{2}{\chi_{\alpha}^{2}(2r+2)}\cdot\sum_{i=1}^{n}t_{i}^{\beta} \text{ for }r\ \geq\ 1\] (A4)  
where there are no failures, we know that \(2 ln(\alpha^{-1})=\chi_{2}^{2}(2)\). That is, at this moment, the first term, \(ln\frac{1}{\alpha}\), is mathematically identical to the Chi-squared value, \(\frac{\chi_{\alpha}^{2}(2)}{2}\), if \(p\)-value is \(\alpha\). The characteristic life, \(\eta_{\alpha}\), could be defined as follows:  
\[\eta_{\alpha}^{\beta}=\frac{2}{\chi_{\alpha}^{2}(2)}\cdot\sum_{i=1}^{n}t_{i}^ {\beta}=\frac{1}{ln\frac{1}{\alpha}}\cdot\sum_{i=1}^{n}t_{i}^{\beta}\text{ for }r=0\] (A5)  
Thus, Equation (A4) can be applicable to all occasions. In other words,  
\[\eta_{\alpha}^{\beta}=\frac{2}{\chi_{\alpha}^{2}(2r+2)}\cdot\sum_{i=1}^{n}t_{ i}^{\beta}\text{ for }r\ \geq\ 0\] (A6)  
If Equation (A6) is inserted into Equation (A2), BX life may be expressed as  
\[L_{B}^{\beta}=\left(ln\frac{1}{1-x}\right)\cdot\eta^{\beta}\cong x\cdot\eta^ {\beta}=x\cdot\frac{2}{\chi_{\alpha}^{2}(2r+2)}\sum_{i=1}^{n}t_{i}^{\beta} \text{ for }x\leq\ 0.2\ (20\%)\] (A7)  
If the sample size is big enough, the planned testing time might advance as  
\[\sum_{i=1}^{n}t_{i}^{\beta}\cong n\cdot h^{\beta}\] (A8)  
where \(h\) is the planned test time  
The approximated lifetime (\(L_{B}\)) in the test might be as lengthy as the targeted life (\(L_{B}^{*}\)).