Module 3 : Microscopic Traffic Flow Modeling
Lecture 13 : Vehicle Arrival Models: Count
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Numerical Example

Compute the $ X^2$ statistic of the following distribution, where $ N=2434$.
Table 1: $ \chi ^2$distribution
$ h$ $ h+dh$ $ p_i^o$ $ p_i^c$
0.0 1 0.012 0.249
1.0 2 0.178 0.187
2.0 3 0.316 0.140
3.0 4 0.218 0.105
4.0 5 0.108 0.079
5.0 6 0.055 0.060
6.0 7 0.033 0.045
7.0 8 0.022 0.034
8.0 9 0.013 0.025
9.0 $ >$ 0.045 0.076
Total   1 1

Solution:

The given headway range and the observed probability is given in column (2), (3) and (4). The observed frequency for the first interval (0 to 1) can be computed as the product of observed probability $ p_i$ and the number of observation (N) i.e. $ f_i^o = p_i^o\times N = 0.012\times2434 = 29.21$ as shown in column (5). Now the computed frequency for the first interval (0 to 1) is the product of computed probability and the number of observation (N) i.e. $ f_i^c = p_i^c\times N = 0.249\times2434 = 441.21$ as shown in column (7). The $ \chi ^2$ value can be computed as $ \frac{(29.21-441.21)^2}{441.21}=384.73$. Similarly, all the rows are computed and the total $ \chi ^2$ value is obtained as 1825.52. A chi-square table gives $ X^2$ values for various degree of freedom. The degree of freedom (DOF) is given as: $ DOF = n-1-p = 10-1-1=8$, where n is the number of intervals (10), and p is the number of parameter (1 because it is exponential distribution). Now at a significance level of 0.05 and DOF 8, from the table, $ X^2_T = 15.5$. Since $ \chi^2_T < \chi^2_C$ hence reject that the observed frequency follows exponential distribution.
Table 2: Solution using comparison with $ X^2$
$ No$ $ h$ $ h+dh$ $ p_i^o$ $ f_i^o$ $ p_i^c$ $ f_i^c$ $ \chi ^2$
(1) (2) (3) (4) (5) (6) (7) (8)
1 0.0 1 0.012 29.21 0.249 441.21 384.73
2 1.0 2 0.178 433.25 0.187 361.23 14.36
3 2.0 3 0.316 769.14 0.140 295.75 757.73
4 3.0 4 0.218 530.61 0.105 242.14 343.67
5 4.0 5 0.108 262.87 0.079 198.25 21.07
6 5.0 6 0.055 133.87 0.060 162.31 4.98
7 6.0 7 0.033 80.32 0.045 132.89 20.79
8 7.0 8 0.022 53.55 0.034 108.80 28.06
9 8.0 9 0.013 31.64 0.025 89.08 37.03
10 9.0 $ >$ 0.045 109.53 0.076 402.34 213.10
  Total   1   1   1825.52