## 5 Matching Analysis of Modelled Parameters
### Parameters' Calculation and Analysis
#### 5.2.2 Braking Distance and Average Braking Deceleration  
Based on the target braking deceleration data shown in Figure 2, the braking distances for high-speed trains, at all levels of braking initial speed and under typical braking conditions, consisting of no line slope, train loads of AW_2, and the three braking modes, could be calculated as Equation (6). Results are shown in Table 6.  
The data in Table 6 were tested separately for normality, and the braking distance data of the carbon-ceramic group significantly obeyed the normal distribution with N\({}_{1}\) (3217, 21852) at the confidence level of 0.05; that of the cast-steel group significantly obeyed the normal distribution with N\({}_{2}\) (4335, 30282)1 at the confidence level of 0.05. A statistical hypothesis test was performed on both groups of data to verify that at the confidence level of 0.05, trains equipped with a carbon-ceramic friction sub could reach a 20% reduction in braking distance compared with trains equipped with a cast-steel friction sub. The carbon-ceramic group braking distance data was set as sample 1, and a new group of data assumed to obey the normal distribution with the N\({}_{3}\) (3468, 30282) was set as sample 3, whose average value is 80% of the cast-steel group braking distance data set as sample 2. Then, to verify the hypothesis, the data from sample 1 and sample 3 were compared to prove that: \(H_{0}:\ \mu_{1}\leq\mu_{3}\left(H_{1}:\ \mu_{1}>\mu_{3}\right).\)  
Such questions are categorised in statistics as mean tests for multiple-sample functional data [24]. The rejection domain of the mean test is constructed as shown in Equation (8), since both groups of samples obey different normal distribution and each variance is known. \[W_{1}=\left\{(x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}):U=\frac{\overline{x}- \overline{y}}{\sqrt{\frac{\epsilon_{1}^{2}}{m}+\frac{\epsilon_{2}^{2}}{2}}}> \mu_{1-\alpha}\right\} \tag{8}\]  
Here, \(\overline{x}\), \(\overline{y}\) are the means of the two groups of samples; \(\sigma_{1}^{2},\sigma_{2}^{2}\) are the variances; \(m,n\) are the data amounts of the two groups of samples; \(\mu_{1-\alpha}\) denotes the standard normal distribution; and the confidence level \(\alpha\) is 0.05. Taking the relevant data of sample 1 and sample 3 into Equation (8), the calculated value of \(U\) equals 0.285, which is smaller than \(\mu_{0.95}\), which equals 1.645. The rejection domain is not satisfied. This means that the \(H_{0}\) hypothesis is accepted, which proves that the mean value of sample 1 is smaller than that of sample 3. Then, the conclusion that the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & **Value of Distance** & & & **Unit** \\ \hline \multicolumn{2}{c}{**Initial Speed**} & **160** & **200** & **250** & **300** & **350** & **380** & **Average** & **km/h** \\ \hline \multirow{4}{*}{\begin{tabular}{c} Carbon-ceramic group \\ \end{tabular} } & Emergency braking & 894 & 1366 & 2085 & 3017 & 4300 & 6068 & \\ \cline{2-9}  & \begin{tabular}{c} Segmented \\ emergency braking \\ \end{tabular} & 844 & 1329 & 2072 & 3051 & 4368 & 6151 & \\ \cline{2-9}  & \begin{tabular}{c} Maximum common \\ braking \\ \end{tabular} & 1013 & 1546 & 2360 & 3504 & 5457 & 8483 & \\ \hline \multirow{4}{*}{\begin{tabular}{c} Cast-steel group \\ \end{tabular} } & Emergency braking & 1092 & 1668 & 2547 & 3718 & 5445 & 8569 & \\ \cline{2-9}  & \
\begin{tabular}{c} Segmented \\ emergency braking \\ \end{tabular} & 1046 & 1603 & 2452 & 3732 & 6145 & 9186 & \\ \cline{2-9}  &
\begin{tabular}{c} Maximum common \\ braking \\ \end{tabular} & 1272 & 2020 & 3499 & 5591 & 8271 & 10,173 & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Braking distance data for braking simulation. trains equipped with a carbon-ceramic friction sub can shorten the braking distance by 20% compared with those equipped with a cast-steel friction sub can be accepted.