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)	26	5	21	37	22	23	18	5
y (gal)	45	3	34	53	33	37	28	9

1. Verify that Σx = 157, Σx² = 3873, Σy = 242, Σy² = 9322, and Σxy = 5986.

(A) Compute r =_______

The data in part 1 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	23	41	16	26	2	37
y (gal)	65	8	15	50	25	33	4	71

2. Compute x and y for both sets of data pairs and compare the averages.

	x	y
Data 1
Data 2

3. 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

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

(A) The standard deviations for x and y are larger for the first set of data.

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

(C) The standard deviations for x and y are the same for both sets of data.

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

(A) Dividing by smaller numbers results in a larger value.

(B) Multiplying by smaller numbers results in a smaller value.   

(C) Multiplying by smaller numbers results in a larger value.

(D) Dividing by smaller numbers results in a smaller value.

6. Verify that Σx = 171, Σx² = 5015, Σy = 271, Σy² = 13,785, and Σxy = 7750.

Compute r= ____


7. Compare r from part 1 with r from part 2. Do the data for averages have a higher correlation coefficient than the data for individual measurements?

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

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

8. 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

1)

X	Y	XY	X²	Y²
26	45	1170	676	2025
5	3	15	25	9
21	34	714	441	1156
37	53	1961	1369	2809
22	33	726	484	1089
23	37	851	529	1369
18	28	504	324	784
5	9	45	25	81

	X	Y	XY	X²	Y²
total sum	157	242	5986	3873	9322

A)

sample size ,   n =   8      
here, x̅ =Σx/n =   19.625   ,   ȳ = Σy/n =   30.250
              
SSxx =    Σx² - (Σx)²/n =   791.88      
SSxy=   Σxy - (Σx*Σy)/n =   1236.75      
SSyy =    Σy²-(Σy)²/n =   2001.50      

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.982
====================

2)

	x	y
Data 1	19.625	30.25
Data 2	21.375	33.875

3)

	sx	sy
Data 1	10.6360	16.9094
Data 2	13.9380	25.6484

4) (B) The standard deviations for x and y are larger for the second set of data.  

5) (A) Dividing by smaller numbers results in a larger value.

6) r= 0.782

7) (B) Yes, the data for averages have a higher correlation coefficient than the data for individual measurements.