Question

Fuming because you are stuck in traffic? Roadway congestion is a costly item, both in time wasted and fuel wasted. Let xrepresent the average annual hours per person spent in traffic delays and let y represent the average annual gallons of fuel wasted per person in traffic delays. A random sample of eight cities showed the following data.

x (hr)	27	5	22	39	22	22	18	5
y (gal)	47	3	34	57	35	38	25	9

(a) Draw a scatter diagram for the data.

Verify that Σx = 160, Σx² = 4076, Σy = 248, Σy² = 9998, and Σxy = 6356.

Compute r.

The data in part (a) represent average annual hours lost per person and average annual gallons of fuel wasted per person in traffic delays. Suppose that instead of using average data for different cities, you selected one person at random from each city and measured the annual number of hours lost x for that person and the annual gallons of fuel wasted y for the same person.

x (hr)	24	4	18	43	16	27	2	39
y (gal)	63	8	10	51	25	31	4	74

(b) Compute x and y for both sets of data pairs and compare the averages.

	x	y
Data 1
Data 2

Compute the sample standard deviations s_x and s_y for both sets of data pairs and compare the standard deviations.

	sx	sy
Data 1
Data 2

In which set are the standard deviations for x and y larger?

The standard deviations for x and y are larger for the first set of data.The standard deviations for x and y are larger for the second set of data.     The standard deviations for x and y are the same for both sets of data.

Look at the defining formula for r. Why do smaller standard deviations s_x and s_y tend to increase the value of r?

Multiplying by smaller numbers results in a larger value.Dividing by smaller numbers results in a smaller value.     Dividing by smaller numbers results in a larger value.Multiplying by smaller numbers results in a smaller value.

(c) Make a scatter diagram for the second set of data pairs.

Verify that Σx = 173, Σx² = 5275, Σy = 266, Σy² = 13,812, and Σxy = 8048.

Compute r.


(d) Compare r from part (a) with r from part (b). Do the data for averages have a higher correlation coefficient than the data for individual measurements?

Yes, the data for averages have a higher correlation coefficient than the data for individual measurements.No, the data for averages do not have a higher correlation coefficient than the data for individual measurements.     

List some reasons why you think hours lost per individual and fuel wasted per individual might vary more than the same quantities averaged over all the people in a city.

Answer 1

Data 1:

Following is the scatter plot:

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	27	47	729	2209	1269
	5	3	25	9	15
	22	34	484	1156	748
	39	57	1521	3249	2223
	22	35	484	1225	770
	22	38	484	1444	836
	18	25	324	625	450
	5	9	25	81	45
Total	160	248	4076	9998	6356

Sample size: n=8

The coefficient of correlation is :

Following table shows the calculations:

	X	Y	(x-mean)^2	(y-mean)^2
	27	47	49	256
	5	3	225	784
	22	34	4	9
	39	57	361	676
	22	35	4	16
	22	38	4	49
	18	25	4	36
	5	9	225	484
Total	160	248	876	2310

Mean:

The standard deviation:

Data 2:

Following is the scatter plot:

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	24	63	576	3969	1512
	4	8	16	64	32
	18	10	324	100	180
	43	51	1849	2601	2193
	16	25	256	625	400
	27	31	729	961	837
	2	4	4	16	8
	39	74	1521	5476	2886
Total	173	266	5275	13812	8048

The coefficient of correlation is :

Following table shows the calculations:

	X	Y	(x-mean)^2	(y-mean)^2
	24	63	5.640625	885.0625
	4	8	310.640625	637.5625
	18	10	13.140625	540.5625
	43	51	456.890625	315.0625
	16	25	31.640625	68.0625
	27	31	28.890625	5.0625
	2	4	385.140625	855.5625
	39	74	301.890625	1660.5625
Total	173	266	1533.875	4967.5

Mean:

The standard deviation:

The standard deviations for x and y are larger for the second set of data.

Dividing by smaller numbers results in a larger value.

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(d)

Yes, the data for averages have a higher correlation coefficient than the data for individual measurements.