## 5 Results
### Relative Difference in Speed and Spacing between \"without\" and \"with\" Periods  
To evaluate the effectiveness of the in-vehicle warning information, this study investigates the distributions of speed and spacing between the \"without\" and \"with\" periods. Figure 6 and Table 3 summarize the distributions and basic statistics of speed from the \"without\" and \"with\" periods, respectively. As shown in Figure 6 and Table 3, the average speed after the in-vehicle warning is lower than before in both periods. Such differences are expected according to Figure 5. Interestingly, we observed that the standard deviation of speed after the warning was larger than that before in both cases but the differences in \"with\" was much higher than in \"without\". It is suspected to be the result of driver deceleration to obtain additional spacing when the warning information is actually provided, which can also be confirmed in Figure 5.  
In addition, this study conducted t-tests of speed distribution, as illustrated in Figure 6, to confirm the statistical significance. Table 4 shows the one-tailed two sample t-test result in speed for \"without\" and \"with\" periods, which was conducted with a significance level of \(\alpha=0.05\). The results in Table 4 reveal that a significant difference in speed between before and after the in-vehicle warning indicated that average speed before the warning was faster than after in both periods. This study repeats the investigation in Figure 6, Tables 3 and 4 for spacing and the results are summarized in Figure 7, Tables 5 and 6. As shown in Figure 7 and Table 5 for \"without\", the average spacing after the warning was similar to before the warning. In Table 5 for \"with\", on the other hand, the difference between the two seems clear: the average spacing after the warning was lower than before. The contrast between the \"without\" and \"with\" periods in spacing supports the premise \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{\(\mathbf{H_{0}: Speed}_{before}=\) Speed\({}_{after}\)} & & & \\  & & \(\mathbf{H_{A}: Speed}_{before}<\) Speed\({}_{after}\) & & & & \\ \hline \multirow{2}{*}{**Speed**} & \multicolumn{2}{c}{**Mean**} & \multicolumn{2}{c}{**Standard Deviation**} & \multirow{2}{*}{**t**} & \multicolumn{2}{c}{\(p\)**-Value**} \\ \cline{2-2} \cline{5-6}  & **Before** & **After** & **Before** & **After** & & **(\({}^{*}p<0.05\))** \\ \hline “without” & 76.47 & 74.46 & 13.54 & 15.43 & \(-\)6.9966 & 0.000 \({}^{*}\) \\ “with” & 79.23 & 70.75 & 10.11 & 18.18 & \(-\)70.2528 & 0.000 \({}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: T-test result in speed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Without**} & \multicolumn{2}{c}{**With**} \\ \cline{2-5}
**Statistics for Spacing** & **Before** & **After** & **Before** & **After** \\ \hline Number of records & 5048 & 5139 & 31,013 & 26,161 \\ Mean (m) & 56.81 & 55.52 & 41.36 & 52.84 \\ Maximum (m) & 138.00 & 137.00 & 138 & 143 \\ Minimum (m) & 3.00 & 3.00 & 3.00 & 2.00 \\ Standard deviation (m) & 28.77 & 28.24 & 33.13 & 28.39 \\ Variance (m\({}^{2}\)) & 827.72 & 797.70 & 1097.64 & 805.73 \\ Median (m) & 56.00 & 56.00 & 40.00 & 52.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistics of “without” and “with” for spacing. Figure 7: Spacing distribution of (**a**) “without” and (**b**) “with”. that the in-vehicle warning induces drivers to take additional spacing at a given speed. Table 6 summarizes the one-tailed two sample t-test result in spacing for both periods. According to the \(p\)-value during the \"without\" period, we cannot reject the null hypothesis that spacing before the in-vehicle warning is the same as after while we accept the alternative hypothesis during the \"with\" period. Summarizing the investigation in speed and spacing between the \"with\" and \"without\" periods, we saw that average speed after the in-vehicle warning was lower than before during both periods.