## 4 Case Study
Various tests are available for assessing the fit quality between the data and the theoretical distribution. In this study, the Kolmogorov-Smirnov (K-S) test was chosen due to its simplicity and reliable results, especially for data sets with limited data. \begin{table}
\begin{tabular}{c c c c} \hline Bearings & & \(\beta\) & \\ \hline
1 & 5.67 & 8.34 & 14.34 \\
2 & 4.15 & 7.22 & 12.75 \\
3 & 7.19 & 9.98 & 15.11 \\
4 & 3.91 & 6.97 & 11.86 \\
5 & 2.77 & 5.74 & 10.78 \\ \hline Bearings & & \(\eta\) & \\ \hline
1 & 544.75 & 584.81 & 626.10 \\
2 & 476.28 & 509.95 & 551.36 \\
3 & 579.78 & 601.29 & 631.04 \\
4 & 493.96 & 529.86 & 572.89 \\
5 & 611.07 & 656.96 & 701.03 \\ \hline \end{tabular}
\end{table} 
Table 1: The expected value for \(\beta\) and \(\eta\) for the bearings of five centrifugal pumps obtained by EM algorithm and respective confidence interval obtained by the bootstrap-T method. The K-S test involves three steps:  
1. Formulation of hypotheses: The null hypothesis (\(H_{0}\)) states that the population from which the data are derived follows the Weibull distribution. The alternative hypothesis (\(H_{1}\)) suggests that the population does not follow the Weibull distribution. 2. Determination of the D value: The D value is calculated as the maximum absolute difference between the sample distribution function F(\(t_{i}\)) and the population distribution function F(\(t\)). This value is compared to the critical value, which depends on the sample size (\(n\)) and the chosen significance level (\(\alpha\)). Additionally, the critical value is adjusted based on the shape parameter value (\(\beta\)): K = 0.70 (\(\beta\) > 3.0); K = 0.75 (1.5 < \(\beta\) < 3); K = 0.8 (\(\beta\) < 1.5). 3. Comparison: If the test statistic D is greater than or equal to the corrected critical value (D \(\geq\) k x critical value), the null hypothesis is rejected (Reject \(H_{0}\)). The \(p\)-value also provides insight into the quality of the fit. The \(p\)-value represents the probability of observing results as extreme as those obtained if the null hypothesis is true. A large \(p\)-value supports \(H_{0}\), while a small \(p\)-value indicates evidence against \(H_{0}\). If the \(p\)-value is greater than 0.05, there is no evidence against \(H_{0}\). For bearing 1, with a sample size of \(n\) = 6 and a significance level of \(\alpha\) = 5%, the following results were obtained: D = 0.218 < 0.363 (0.70 \(\times\) 0.519); \(p\)-value = 0.5294 > 0.05. Based on these results, the null hypothesis is not rejected, indicating that the population from which the data are derived follows the Weibull distribution. Similar results were obtained for the other bearings. Estimating the parameters of the Weibull distribution allows for comprehensive reliability analysis of industrial equipment, specifically in centrifugal pumps. By understanding failure patterns through parameter estimation, system designs can be optimized, appropriate materials can be selected, and maintenance intervals can be determined. Reliability estimates derived from Weibull analysis serve as a quantitative basis for decision making, facilitating efficient resource allocation, reducing downtime, and improving system performance. Furthermore, Weibull analysis enables comparative failure data analysis from different systems, components, or designs. The relative reliability of various products or configurations can be assessed by comparing the estimated parameters.