## 3 Results
### Experiment 1: Assisted FRT Performance
#### 3.1.2 Statistical Analysis  
The outcomes of Shapiro-Wilk normality test (test statistic W and \(p\)-value) on the data obtained manually using the whiteboard (M_FRT) and by the system using Kinect (K_FRT) are shown in Table 1. The \(p\)-value of the data in both methods (M_FRT) and (K_FRT) is greater than 0.05 then the null hypothesis is maintained. Thus, we can assume that our data are approximately normally distributed. Figure 7: Box plot of the FRT measurements by the experimental system (K_FRT) and manually (M_FRT) in Experiment 1. Visual inspection using normal Q-Q plots indicates also that the data obtained by both methods are approximately normally distributed as it can be seen in Figure 8 where the dots are approximately distributed along the line because they are located in the 95% confidence bands for the empirical quantile plot [46]. To see the difference between the means of the two sets of data, we used the paired \(t\)-test. As shown in Table 2, the \(p\)-value is much less than 0.05 so the null hypothesis is rejected and there is statistically significance difference in average means of the measures obtained by the two methods, which is not correct since we know from previous work [26] that there are no statistically significant differences in the average, so the measurements are valid but need to be corrected. The Pearson correlation coefficient obtained is equal to 0.97 with \(p\)-value less than the significance level 0.05 (\(p\)-value = \(1.534\times 10^{-12}\)). Hence, we can conclude that FRT values obtained by the Microsoft Kinect v2 and those measured manually are significantly correlated. This can also be shown visually in Figure 9 where the scatter plot shows linear correlation between the data. Moreover, we found a good repeatability of the FRT measurements tracked by Kinect with an ICC = 0.861 (95% confidence interval, CL, 0.73-0.939). Given that we demonstrated that the data is linearly correlated, we made a linear model to correct the K_FRT data. Regression results gave a slope coefficient of 0.934 and adjusted R-squared value of 0.99 which means that the regression model explains 99% of the variability in the observations. We also found that the residuals are normal by applying Shapiro-Wilk normality test (W = 0.90586, \(p\)-value = 0.06218). In order to inspect visually the linear regression assumptions, we examined the residual plots shown in Figure 10. The first is a constant variance plot, which checks for the homogeneity of the variance and the linear relation. There is no obvious pattern in this graph, then linearity assumption is met. The second plot in the graph is a Q-Q plot, which checks if residuals are normally distributed. The points fall on a line hence the normality assumption is also met. The third plot allows to detect heterogeneity of the variance while the fourth plot allows for the detection of points that have a large impact on the regression coefficients. Visual inspection to these plots show that the assumption of equal variance (homoscedasticity) is met and \begin{table}
\begin{tabular}{l l} \hline \hline  & **M\_FRT-K\_FRT** \\ \hline W & 0.93984 \\ \(p\)-value & 0.262 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Shapiro–Wilk normality test applied to data of Experiment 1. \begin{table}
\begin{tabular}{l} \hline \hline t = \(-\)6.2064 \\ \hline \(p\)-value = 7.405 \(\times\) 10\({}^{-6}\) \\ \hline Difference = \(-\)2.637895 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of paired \(t\)-test applied to data of Experiment 1. Figure 8: Q-Q Plot of data of Experiment 1 (a) M_FRT and (b) K_FRT. there is no influential cases.