Chapter 4, Problem 9P

To determine

Whether there was a statistically significant difference in the average speeds, the mean speed, standard deviation, 85^th percentile speed and percentage of traffic exceeding the posted speed limit of 30 mi/h.

Answer to Problem 9P

Significant reduction

  ${\overline{u}}_1=34.9mi/h$, ${\overline{u}}_2=27.5mi/h$, $S_1=\pm 3.5mi/h$, $S_2=\pm 5.3mi/h$, $v_{{85}_1}=36mi/h$, $v_{{85}_2}=31.5mi/h$, 76 %, 24 %

Explanation of Solution

Given:

Significance level of $\alpha =0.05$

Formula used:

  $\overline{u}=\frac{{\displaystyle \sum f_i{u}_i}}{{\displaystyle \sum f_i}}$

  $\overline{u}$ is arithmetic mean

  $f_i$ is number of observations in each speed group

  $u_i$ is mid-value for the ith speed group

  $S=\sqrt{\frac{{\displaystyle \sum f_i{\left(u_i-\overline{u}\right)}^2}}{N-1}}$

S is standard deviation

N is number of observations

  $S_p{}^2=\frac{\left(n_1-1\right)S_1{}^2+\left(n_2-1\right)S_2{}^2}{n_1+n_2-2}$.

S_pis square root of the pooled variance

S₁ and S₂ are standard deviations of the populations

  $n_1$ and $n_2$ are sample means

  $T=\frac{{\overline{u}}_1-{\overline{u}}_2}{S_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

T is the test static

  ${\overline{u}}_1$ and ${\overline{u}}_2$ are sample means

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/hgiving a speed range of 12. For five classes, the range per class is 2.4mi/h. A frequency distribution table can then be prepared, as shown below, in which the speed classes are listed in column 1 and the mid-values are in column 2. The number of observationsfor each class is listed in column 3, the cumulative percentages of all observations arelisted in column 6.

1	2	3	4	5	6	7
Speed class (mi/h)	Class mid-value   $u_i$	Class frequency,   $f_i$	$f_i{u}_i$	Percentage of class frequency	Cumulative percentage of class frequency	$f_i{\left(u_i-\overline{u}\right)}^2$
28-30	29	4	116	13	13	139.24
31-33	32	5	160	17	30	42.05
34-36	35	12	420	40	70	0.12
37-39	38	6	228	20	90	57.66
40-42	41	3	123	10	100	111.63
Total		30	1047			350.7

Determine the arithmetic mean speed:

  $\begin{array}{l}\overline{u}=\frac{{\displaystyle \sum f_i{u}_i}}{{\displaystyle \sum f_i}}\\ {\displaystyle \sum f_i=30}\\ {\displaystyle \sum f_i{u}_i}=1047\\ {\overline{u}}_1=\frac{1047}{30}=34.9mi/h\end{array}$

Determine the standard deviation:

  $\begin{array}{l}S=\sqrt{\frac{{\displaystyle \sum f_i{\left(u_i-\overline{u}\right)}^2}}{N-1}}\\ {\displaystyle \sum f_i{\left(u_i-\overline{u}\right)}^2}=350.7\\ N-1={\displaystyle \sum f_i-1=30}-1=29\\ S_1=\sqrt{\frac{350.7}{29}}\\ S_1=\pm 3.5mi/h\end{array}$

The 85th-percentile speed is obtained from the cumulative frequency distribution curve as 36 mi/h.

The percentage of traffic exceeding the posted speed limit of 30 mi/h is 76 %.

Below given figure shows the cumulative frequency distribution curve for the data given. In this case, the cumulative percentages in column 6 of the above Table are plotted against the upper limit of each corresponding speed class. This curve, therefore, gives the percentage of vehicles that are traveling at or below a given speed.

After an increase in speed enforcement activities:

The speed ranges from 20 to 37 mi/h giving a speed range of 17. For six classes, the range per class is 2.83 mi/h. A frequency distribution table can then be prepared, as shown below in which the speed classes are listed in column 1 and the mid-values are in column 2. The number of observations for each class is listed in column 3, the cumulative percentages of all observations are listed in column 6.

1	2	3	4	5	6	7
Speed class (mi/h)	Class mid-value   $u_i$	Class frequency,   $f_i$	$f_i{u}_i$	Percentage of class frequency	Cumulative percentage of class frequency	$f_i{\left(u_i-\overline{u}\right)}^2$
20-22	21	6	126	20	20	253.5
23-25	24	8	192	27	47	98
26-28	27	4	108	13	60	1
29-31	30	3	90	10	70	18.75
32-34	33	5	165	17	87	151.25
35-37	36	4	144	13	100	289
Total		30	825			811.5

Determine the arithmetic mean speed:

  $\begin{array}{l}\overline{u}=\frac{{\displaystyle \sum f_i{u}_i}}{{\displaystyle \sum f_i}}\\ {\displaystyle \sum f_i=30}\\ {\displaystyle \sum f_i{u}_i}=825\\ {\overline{u}}_2=\frac{825}{30}=27.5mi/h\end{array}$

Determine the standard deviation:

  $\begin{array}{l}S=\sqrt{\frac{{\displaystyle \sum f_i{\left(u_i-\overline{u}\right)}^2}}{N-1}}\\ {\displaystyle \sum f_i{\left(u_i-\overline{u}\right)}^2}=811.5\\ N-1={\displaystyle \sum f_i-1=30}-1=29\\ S_2=\sqrt{\frac{811.5}{29}}\\ S_2=\pm 5.3mi/h\end{array}$

Below given figure shows the cumulative frequency distribution curve for the data given. In this case, the cumulative percentages in column 6 of the above Table are plotted against the upper limit of each corresponding speed class. This curve, therefore, gives the percentage of vehicles that are traveling at or below a given speed.

The 85th-percentile speed is obtained from the cumulative frequency distribution curve as 31.5 mi/h.

The percentage of traffic exceeding the posted speed limit of 30 mi/h is 24 %.

Determine square root of the pooled variance:

  $\begin{array}{l}{S}_p{}^2=\frac{\left(n_1-1\right)S_1{}^2+\left(n_2-1\right)S_2{}^2}{n_1+n_2-2}\\ n_1=n_2=N=30\\ S_1=3.5\\ S_2=5.3\\ S_p{}^2=\frac{\left(30-1\right){3.5}^2+\left(30-1\right){5.3}^2}{30+30-2}\\ S_p{}^2=20.17\\ S_p=4.49\end{array}$

Compute test static T:

  $\begin{array}{l}T=\frac{{\overline{u}}_1-{\overline{u}}_2}{S_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ {\overline{u}}_1=35.1mi/h\\ {\overline{u}}_2=27.47mi/h\\ T=\frac{35.1-27.47}{4.49\sqrt{\frac{1}{30}+\frac{1}{30}}}=6.58\end{array}$

Determine whether ${\overline{u}}_1$ is significantly higher than ${\overline{u}}_2$ :

  $\text{Degree of freedom = 30 + 30 - 2 = 58}$

From Appendix A, theoretical ${\text{t}}_{\text{α}}\text{= 1}\text{.672}$ for $\alpha \text{=}\text{0}\text{.05}$

Since $6.58>1.672$, then ${\text{T > t}}_{\text{α}}$ and the null hypothesis should be rejected.

Conclusion:

The increase in speed enforcement activities has resulted in a significant reduction in the mean speed on the street at a significance level of 0.05. The mean speeds before and after increase in speed enforcement activities are 35.1 and 27.47 mi/h respectively. The standard deviations are 3.5 and 5.3 mi/h respectively. The 85^th percentile speeds are 36 mi/h and 31.5 mi/h and percentages of traffic exceeding the posted speed limit of 30 mi/h are 76 % and 24 %.