Correlation of Averages Fuming because you are stuck in traffic? Roadway congestion is a costly item, in both 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

(Reference: Statistical Abstract of the United States, 122nd edition).

x (hr) 28 5 20 35 20 23 18 5 y (gal) 48 3 34 55 34 38 28 9

(a) Draw a scatter diagram for the data. Verify that Sx 5154, Sx2 5 3712, Sy 5 249, Sy2 5 9959, and Sxy 5 6067. 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) 20 4 18 42 15 25 2 35 y (gal) 60 8 12 50 21 30 4 70

(b) Compute x and y for both sets of data pairs and compare the averages. Compute the sample standard deviations sx and sy for both sets of data pairs and compare the standard deviations.

In which set are the standard deviations for x and y larger? Look at the defining formula for r, Equation 1. Why do smaller standard deviations sx and sy tend to increase the value of r?

(c) Make a scatter diagram for the second set of data pairs. Verify that Sx 5161, Sx2 5 4583, Sy 5 255, Sy2 512,565, and Sxy 5 7071.

Compute r.

(d) Compare r from part

(a) with r from part

(c). Do the data for averages 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.

AppendixLO1