## 5 Discussion on Statistical Analysis
### Goodness-of-Fit Test  
It is well-known that normal distribution, log-normal distribution, exponential distribution, Weibull distribution, and gamma distribution are commonly used for data analysis of insulating materials. In this paper, the AC-BDS values of epoxy/Al2O3 micro-composites were analyzed using four distributions (Weibull, log-normal, exponential, and gamma distributions), and the goodness-of-fit was assessed for each distribution based on \(p\)-value \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Group**} & \multicolumn{2}{c}{**Relative Permittivity**} & \multicolumn{2}{c}{**Dissipation Factor**} \\ \cline{2-5}  & **Statistics \(t\)** & \(p\)**-Value** & **Statistics \(t\)** & \(p\)**-Value** \\ \hline A & 5.7674 & 2.7 \(\times\)\(10^{-4}\) & 4.3244 & \(1.9\times\)\(10^{-3}\) \\ B & 10.661 & 2.1 \(\times\)\(10^{-5}\) & 7.3442 & \(4.4\times\)\(10^{-5}\) \\ C & 10.556 & 2.3 \(\times\)\(10^{-6}\) & 8.8547 & \(9.8\times\)\(10^{-6}\) \\ D & 8.7395 & 1.1 \(\times\)\(10^{-5}\) & 11.274 & \(1.3\times\)\(10^{-6}\) \\ E & 7.0683 & 5.9 \(\times\)\(10^{-5}\) & 8.6363 & \(1.2\times\)\(10^{-5}\) \\ F & 7.8335 & 2.6 \(\times\)\(10^{-5}\) & 7.8994 & \(2.5\times\)\(10^{-5}\) \\ G & 8.3325 & 1.6 \(\times\)\(10^{-5}\) & 10.729 & \(2.0\times\)\(10^{-6}\) \\ H & 5.1203 & 6.3 \(\times\)\(10^{-4}\) & 6.6813 & \(9.0\times\)\(10^{-5}\) \\ I & 5.9924 & 2.0 \(\times\)\(10^{-4}\) & 10.257 & \(2.9\times\)\(10^{-6}\) \\ J & 4.4111 & 1.7 \(\times\)\(10^{-3}\) & 6.122 & \(1.8\times\)\(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Application of paired \(t\)-test regarding dielectric properties with different temperatures. The A-D statistic is an indicator that allows the analyst to determine how well the observed data (in this case, AC-BDS) fit a given distribution. In the analysis of the RT-BDS values shown in Table 9, the log-normal distribution had the smallest value of 0.533, while the exponential distribution had the largest value of 8.403. The A-D values for the Weibull and gamma distribution were found to be 1.124 and 0.547, respectively. Similarly, in the analysis of the ST-BDS values presented in Table 10, the log-normal distribution again had the smallest A-D value of 0.181, while the exponential distribution had the largest value of 8.566. The Weibull distribution was 0.037 and the gamma distribution was 0.013 larger than the log-normal distribution. From these results, it can be concluded that the log-normal distribution is the best distribution for analyzing the AC-BDS values of epoxy/Al\({}_{2}\)O\({}_{3}\) micro-composites specimens. In this research, the A-D goodness-of-fit test results showed that the log-normal distribution is the best fit for the experiment data of this study. While normal and Weibull distributions have been widely applied to data analysis in high-voltage experiments [24], the log-normal distribution has been considered as a suitable choice for characterizing -voltage experiment data in statistical standard [23] and relevant studies [17,50]. Meanwhile, the exponential distribution showed considerably larger A-D statistics compared to other statistical distributions. In fact, unlike other distributions, the exponential distribution corresponds to the constant failure rate (CFR) in the bathtub curve, while the other distributions correspond to the increasing failure rate (IFR). The exponential distribution corresponds to the CFR because it reflects the characteristic that the failure rate does not change over time during the useful life of the product [51]. In other words, the exponential distribution is suitable when the failure rate remains constant over time, regardless of the duration. Failures due to deterioration are gradual over time, with components exhibiting an increasing failure rate (i.e., IFR) as they are affected by fatigue, wear, stress, etc. Consequently, the exponential distribution model, which assumes a constant failure rate, is not suitable for explaining failures caused by degradation.