## 4 Results
### Comparison with Experiments and Finite Element Simulations  
As depicted in Figure 10, the comparison between experimental data and FEA results for compressive residual stress values at different positions reveals a close correlation between the two sets of data. The experimental and FEA results for each position were subjected to a paired \(t\)-test to assess the consistency and reliability of the FEA model in predicting the residual stress distribution. The hypothesis test was designed to evaluate whether there is a statistically significant difference between the experimental and simulated values. The null hypothesis (H\({}_{0}\)) states that there is no difference between the mean compressive residual stress values obtained from the experiments and FEA, while  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample** & **N** & **Mean** & **Stdev** & **SE Mean** & **T-Value** & _p-Value_ \\ \hline Experimental & 7 & 498.9 & 148.3 & 56.1 & & \\ FEA & 7 & 502.5 & 158.8 & 60.0 & \(-\)0.22 & 0.833 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Paired \(t\)-test Results.  
Figure 9: FEA of residual stress at specific points in a deep-rolled surface.  
the alternative hypothesis (H\({}_{1}\)) asserts that there is a significant difference between the two. In the analysis, seven sample points (P1 to P7) were evaluated, yielding a mean compressive residual stress of 498.9 MPa for the experimental results and 502.5 MPa for the FEA simulations. The standard deviations for the experimental and FEA results were 148.3 MPa and 158.8 MPa, respectively, indicating a similar spread in the data for both methods. Table 7 shows a t-value of \(-0.22\), coupled with a \(p\)-value of 0.833, indicating that the difference between the means is not statistically significant. Since the \(p\)-value is greater than the conventional significance level of 0.05, the null hypothesis cannot be rejected. This suggests that the variations observed between the experimental and FEA results are due to random fluctuations rather than inherent discrepancies between the two methods. Overall, the results indicate that the FEA model provides a reliable prediction of compressive residual stress when compared to experimental data.  
The minor differences in stress values between the two methods do not indicate any systematic errors in the FEA approach. Therefore, the FEA model can be considered a valid tool for predicting the residual stress distribution in similar deep rolling processes, as it demonstrates consistency with experimentally measured values. This consistency is critical for validating the FEA as a useful simulation tool in engineering applications.  
In Figure 11, the boxplot illustrates the differences between the experimental and finite element analysis (FEA) compressive residual stress values, with the null hypothesis and a 95% confidence interval for the meaning of the differences. The boxplot's structure displays the interquartile range (IQR), capturing the central 50% of the data points, with the median represented by the line inside the box. The whiskers indicate the range of the data outside the IQR, showing variability in the differences. The red circle labeled \"H\({}_{0}\)\" marks the null hypothesis, where the mean difference is hypothesized to be 0, indicating no significant difference between experimental and FEA results. The mean difference (\(\bar{\triangledown}\)  
Figure 10: Residual stress distribution on the rail surface after the deep rolling process.  
is shown near 0, and the 95% confidence interval for the mean difference also includes 0, suggesting that the difference between the two datasets is not statistically significant. This result implies that the FEA model accurately predicts the residual stresses when compared to experimental data. Given the overlap between the null hypothesis and the confidence interval, we fail to reject the null hypothesis, indicating that the FEA simulation aligns well with the experimental outcomes.