Chapter 4, Problem 10P

To determine

(a)

The histogram frequency distribution, cumulative percentage distribution for each set of data and average speed.

Answer to Problem 10P

${\overline{u}}_1=34.9mi/h$, ${\overline{u}}_2=27.5mi/h$

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

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/h giving a speed range of 12. For five classes, the range per class is 2.4 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 and 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	$f_i{\left(u_i-\overline{u}\right)}^2$
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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

Determine the arithmetic mean speed:

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

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 and 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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

Determine the arithmetic mean speed:

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

Conclusion:

The average speeds of each set of data are 34.9 and 27.5 mi/h respectively.

To determine

(b)

The histogram frequency distribution, cumulative percentage distribution for each set of data and 85^th percentile speed.

Answer to Problem 10P

$v_{{85}_1}=36mi/h$, $v_{{85}_2}=31.5mi/h$

Explanation of Solution

Given:

Significance level of $\alpha =0.05$

  

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/h giving a speed range of 12. For five classes, the range per class is 2.4 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 and 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	$f_i{\left(u_i-\overline{u}\right)}^2$
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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

Below Figure shows the cumulative frequency distribution curve for the data given. In this case, the cumulative percentages in column 6 of above Table are plotted against the upper limit of each corresponding speed class. This curve 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 36 mi/h.

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 and 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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

Below Figure shows the cumulative frequency distribution curve for the data given. In this case, the cumulative percentages in column 6 of above Table are plotted against the upper limit of each corresponding speed class. This curve 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.

Conclusion:

The 85th-percentile speed for each set of data are 36 and 31.5 mi/h respectively.

To determine

(c)

The histogram frequency distribution, cumulative percentage distribution for each set of data and 15^th percentile speed.

Answer to Problem 10P

$v_{{15}_1}=28.5mi/h$, $v_{{15}_2}=0mi/h$

Explanation of Solution

Given:

Significance level of $\alpha =0.05$

  

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/h giving a speed range of 12. For five classes, the range per class is 2.4 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 and 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	$f_i{\left(u_i-\overline{u}\right)}^2$
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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

The 15th-percentile speed is obtained from the cumulative frequency distribution curve as 28.5 mi/h.

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 and 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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

The 15th-percentile speed is obtained from the cumulative frequency distribution curve as 0 mi/h.

Conclusion:

The 15th-percentile speed for each set of data are 28.5 and 0 mi/h respectively.

To determine

(d)

The histogram frequency distribution, cumulative percentage distribution for each set of data and mode.

Answer to Problem 10P

35 mi/h and 24 mi/h

Explanation of Solution

Given:

Significance level of $\alpha =0.05$

  

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/h giving a speed range of 12. For five classes, the range per class is 2.4 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 and 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	$f_i{\left(u_i-\overline{u}\right)}^2$
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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

The mode or modal speed is obtained from the frequency histogram as 35 mi/h

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 and 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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

The mode or modal speed is obtained from the frequency histogram as 24 mi/h.

Conclusion:

The mode for each set of data are 35 and 24 mi/h respectively.

To determine

(e)

The histogram frequency distribution, cumulative percentage distribution for each set of data and median.

Answer to Problem 10P

32.5 and 23.5 mi/h

Explanation of Solution

Given:

Significance level of $\alpha =0.05$

  

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/h giving a speed range of 12. For five classes, the range per class is 2.4 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 and 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	$f_i{\left(u_i-\overline{u}\right)}^2$
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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

The median speed is obtained from the cumulative frequency distribution curve as 32.5 mi/h which is the 50^th percentile 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 and 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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

The median speed is obtained from the cumulative frequency distribution curve as 23.5 mi/h which is the 50^th percentile speed.

Conclusion:

The median speed for each set of data are 32.5 and 23.5 mi/h respectively.

To determine

(f)

The histogram frequency distribution, cumulative percentage distribution for each set of data and pace.

Answer to Problem 10P

32 to 39 mi/h and 27 to 36 mi/h

Explanation of Solution

Given:

Significance level of $\alpha =0.05$

  

Calculation:

Before an increase in speed enforcement activities:

The speed ranges from 28 to 40 mi/h giving a speed range of 12. For five classes, the range per class is 2.4 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 and 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	$f_i{\left(u_i-\overline{u}\right)}^2$
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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

Below figure shows the frequency distribution curve for the data given. In this case, a curve showing percentage of observations against speed is drawn by plotting values from column 5 of above Table against the corresponding values in column 2. The total area under this curve is one or 100 percent.

  

The pace is obtained from the frequency distribution curve above as 32 to 39 mi/h.

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 and 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

Below Figure shows the frequency histogram for the data shown in above Table. The values in columns 2 and 3 of Table are used to draw the frequency histogram, where the abscissa represents the speeds and the ordinate the observed frequency in each class.

  

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

  

Below figure shows the frequency distribution curve for the data given. In this case, a curve showing percentage of observations against speed is drawn by plotting values from column 5 of above Table against the corresponding values in column 2. The total area under this curve is one or 100 percent.

  

The pace is obtained from the frequency distribution curve drawn above as 27 to 36 mi/h.

Conclusion:

The pace for each set of data are 32 to 39 mi/h and 27 to 36 mi/h respectively.