Expand Your Knowledge: Correlation of Averages Fuming

Chapter , Problem 21

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Expand Your Knowledge: Correlation of Averages 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. (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 x 154, x2 3712, y 249, y2 9959, and xy 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 and 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 x 161, x2 4583, y 255, y2 12,565, and xy 7071. 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? 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. x y 0r0