Question

Fuming because you are stuck in traffic? Roadway congestion is a costly item, both in time wasted and fuel wasted. Let x represent 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) 29 5 23 37 18 23 19 5 y (gal) 45 3 31 53 31 38 28 9 (a) Draw a scatter diagram for the data. Submission Data Correct: Your answer is correct. Verify that Σx = 159, Σx2 = 4003, Σy = 238, Σy2 = 9074, and Σxy = 6003. 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) 22 4 19 40 19 26 2 39 y (gal) 63 8 12 54 21 35 4 71 (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 sx and sy 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. Correct: Your answer is correct. Look at the defining formula for r. Why do smaller standard deviations sx and sy tend to increase the value of r? Dividing by smaller numbers results in a smaller value. Multiplying by smaller numbers results in a larger value. Multiplying by smaller numbers results in a smaller value. Dividing by smaller numbers results in a larger value. Correct: Your answer is correct. (c) Make a scatter diagram for the second set of data pairs. Submission Data Correct: Your answer is correct. Verify that Σx = 171, Σx2 = 5023, Σy = 268, Σy2 = 13,816, and Σxy = 7892. 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? No, the data for averages do not 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. Correct: Your answer is correct. 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
	29	45	841	2025	1305
	5	3	25	9	15
	23	31	529	961	713
	37	53	1369	2809	1961
	18	31	324	961	558
	23	38	529	1444	874
	19	28	361	784	532
	5	9	25	81	45
Total	159	238	4003	9074	6003

Sample size: n=8

The coefficient of correlation is :

Following table shows the calculations:

	X	Y	(x-mean)^2	(y-mean)^2
	29	45	83.265625	232.5625
	5	3	221.265625	715.5625
	23	31	9.765625	1.5625
	37	53	293.265625	540.5625
	18	31	3.515625	1.5625
	23	38	9.765625	68.0625
	19	28	0.765625	3.0625
	5	9	221.265625	430.5625
Total	159	238	842.875	1993.5

Mean:

The standard deviation:

Data 2:

Following is the scatter plot:

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	22	63	484	3969	1386
	4	8	16	64	32
	19	12	361	144	228
	40	54	1600	2916	2160
	19	21	361	441	399
	26	35	676	1225	910
	2	4	4	16	8
	39	71	1521	5041	2769
Total	171	268	5023	13816	7892

The coefficient of correlation is :

Following table shows the calculations:

	X	Y	(x-mean)^2	(y-mean)^2
	22	63	0.390625	867.3025
	4	8	301.890625	652.8025
	19	12	5.640625	464.4025
	40	54	346.890625	418.2025
	19	21	5.640625	157.5025
	26	35	21.390625	2.1025
	2	4	375.390625	873.2025
	39	71	310.640625	1402.5025
Total	171	268	1367.875	4838.02

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.