ES9 Additional Exercises

Chapter 3

1. In a study involving children's fear related to being hospitalized, the age and the score each child made on the Child Medical Fear Scale (CMFS) were:

Age (x) 8 9 9 10 11 9 9 9 11 11

CMFS score (y) 31 25 40 27 35 29 25 34 27 36

Construct a scatter diagram of these data. (Retain for use in answering Exercise 3.4.)

2. Walter Payton was one of the NFL's greatest running backs. Below are listed the number of carries and the total yards gained in each of his 13 years with the Chicago Bears.

Number Carries 196 311 339 333 369 317 339 148 314 381 324 321 146

Total Yards 679 1390 1852 1359 1610 1460 1222 596 1421 1684 1551 1333 586

a. Construct a scatter diagram depicting these data.

b. How would you describe this scatter diagram? What do you see that is unusual about the scatter diagram?

c. What circumstances might explain this "two group" appearance of the data points? Explain.

3. Total solar eclipses actually take place nearly as often as total lunar eclipses, but they are visible over a much narrower path. Both the path width and the duration vary substantially from one eclipse to the next. The table below shows the duration (seconds) and path width (miles) of 44 total solar eclipses measured in the past and those projected to the year 2010:

a. Draw a scatter diagram showing duration, y, and path width, x, for the total solar eclipses.

b. How would you describe this diagram?
4. In an article titled "Self-Reported Fears of Hospitalized School-Age Children" (Journal of Pediatric Nursing, Vol. 9, No. 2, 1994), the authors report a correlation of 0.10 between the ages of children and the score they made on the Child Medical Fear Scale (CMFS). Suppose the ages and CMFS scores for ten children were as follows (same data as Exercise 3.1):

CMFS Score 31 25 40 27 35 29 25 34 27 36

Age 8 9 9 10 11 9 9 9 11 11

Find: a. SS(x) b. SS(y) c. SS(xy) d. the value of r for these data

5. Consider the following data, which give the weight (in thousands of pounds) x and gasoline mileage (miles per gallon) y for ten different automobiles.

x 2.5 3.0 4.0 3.5 2.7 4.5 3.8 2.9 5.0 2.2

y 40 43 30 35 42 19 32 39 15 44

Find: a. SS(x) b. SS(y) c. SS(xy) d. Pearson's product moment r

6. An article titled "Leader Power, Commitment Satisfaction, and Propensity to Leave a Job Among U.S. Accountants" (Journal of Social Psychology, Vol. 133, No. 5, Oct. 1993) reported a linear correlation coefficient of -0.61 between satisfaction with work scores and propensity to leave a job scores. Suppose similar assessments of work satisfaction, x, and propensity to leave a job, y, gave the following scores.

x 12 24 17 28 24 36 20

y 44 36 25 23 32 17 24

a. Find the linear correlation between x and y.

b. What does the value of this correlation coefficient seem to be telling us? Explain.

7. Cable television video networks can be measured on the basis of both their number of noncable affiliates and number of subscribers. Below is a table of 18 video networks, their affiliates and subscribers (in millions):

a. Draw a scatter diagram of the two variables, affiliates and subscribers.

b. Find the linear correlation coefficient and interpret the results.

8. An article titled "College Recreation Facility Survey" (Athletic Business, April 1994) reported the following results from 358 four-year colleges and universities in the United States and Canada.

Total Square Feet Devoted

Enrollment Number of Schools to Recreation per School

0 - 1,249 58 47,864

1,250 - 2,499 53 71,828

2,500 - 4,999 53 89,716

5,000 - 9,999 62 101,016

10,000 - 17,999 68 127,952

18,000 or over 64 200,896

a. Using the midpoint of the first five enrollment classes and 25,000 in place of the class 18,000 or over for the x-values and the total square feet devoted to recreation per school for the y-values, find the linear correlation coefficient between x and y.

2. 
   What does the value of this correlation seem to be telling us? Explain.
   

9. An article titled "Microbioluminometry Assay (MBA): Determination of Erythromycin Activity in Plasma or Serum" (Journal of Pharmaceutical sciences, Dec. 1989) compares the MBA assay with another method, agar diffusion plate assay, for determining the erythromycin activity in plasma or serum. The MBA technique is economical because it requires less sample and reagents.

If X represents the agar diffusion plate assay in micrograms per milliliter and Y represents the MBA assay in micrograms per milliliter, then the results may be expressed as

Y = 0.054128 + 0.92012X r = 0.9525 n = 206

If the agar diffusion plate assay determined the level of erythromycin to be 1.200, predict what the MBA assay would be.

10. In the article "Beyond Prediction: The Challenge of Minority Achievement in Higher Education" (Lunneborg and Lunneborg, Journal of Multicultural Counseling and Development), the relationship between high school GPA and first-year university GPA was investigated for the following groups: Asian Americans, blacks, chicanos, Native Americans, and whites. For the 43 Native Americans, the correlation was found to be 0.26 and the equation of the line of best fit was found to be university GPA = 1.85 + 0.30  high school GPA. Both GPAs were 4.0 scales.

Use the equation of the line of best fit to predict the mean first-year-university GPA for all Native Americans who had a high school GPA equal to 3.0.

11. People not only live longer today but also live longer independently. The May/June 1989 issue of Public Health Reports published an article titled "A Multistate Analysis of Active Life Expectancy." Two of the variables studied were a person's current age and the expected number of years remaining.

Age x 65 67 69 71 73 75 77 79 81 83

Years Remaining y 16.5 15.1 13.7 12.4 11.2 10.1 9.0 8.4 7.1 6.4

a. Draw a scatter diagram.

b. Calculate the equation of best fit.

c. Draw the line of best fit on the scatter diagram.

d. What is the expected years remaining for a person who is 70 years old? Find the answer in two different ways: Use the equation from (b) and use the line on the scatter diagram from (c).

e. Are you surprised that the data all lie so close to the line of best fit? Explain why the ordered pairs follow the line of best fit so closely.

12. A record of maintenance costs is kept for each of several cash registers throughout a department store chain. A sample of 14 registers gave the following data:

(years) (dollars) | (years) (dollars)

6 142 | 2 99

7 231 | 1 114

1 73 | 9 191

3 90 | 3 160

6 176 | 8 155

4 132 | 9 231

5 167 | 8 202

a. Draw a scatter diagram that shows these data.

b. Calculate the equation of the line of best fit.

c. A particular cash register is eight years old. How much maintenance (cost) do you predict it will require this year?

d. Interpret your answer to (c).

13. The following data are the ages and the asking prices for 19 used foreign compact cars:

Age x Price y | Age x Price y

(years) (x $100) | (years) (x $100)

3 68 | 6 42

5 52 | 8 22

3 63 | 5 50

6 24 | 6 36

4 60 | 5 46

4 60 | 7 36

6 28 | 4 48

7 36 | 7 20

2 68 | 5 36

2 64 |

b. Calculate the equation of the line of best fit.

c. Graph the line of best fit on the scatter diagram.

d. Predict the average asking price for all such foreign cars that are five years old. Obtain this answer in two ways: Use the equation from (b) and use the line drawn in (c).

14. An article titled "Women, Work and Well-Being: The Importance of Work Conditions (Health and Social Behavior, Vol. 35, No. 3, Sept. 1994) studied 202 full-time homemakers and 197 employed wives. The linear correlation coefficient between family income and education was reported to equal 0.43 for the participants in the study. A similar study involving eight individuals gave the following results. (x represents the years of education, and y represents the family income in thousands of dollars).

x 12 13 10 14 11 14 16 16

y 34 45 36 47 43 35 50 42

a. Find the linear correlation between x and y.

b. Find the equation of the line of best fit.

15. You have probably thought about the number of hours you spend at work or at leisure, but have you ever wondered about the relationship between the number of hours that all Americans spend per week at work and at leisure?

Year	Work	Leisure
1995	50.6	19.2
1993	50.0	19.0
1989	48.7	16.6
1984	47.3	16.6
1980	46.9	19.2
1975	43.1	24.3

a. How many hours did you, personally, spend working last week? How many hours did you spend at leisure pursuits?

b. Construct one graph using years as a horizontal axis, hours as the vertical axis and plot hours of work and hours of leisure using two different colored dots. Connect the dots.

c. Do you see any pattern(s) to the graph? Explain.

d. Is there a relationship between the two variables, hours spent at work and hours devoted to leisure? When people spend more time at one, do they spend less time at the other? Use the techniques learned in this chapter to present graphic and numerical statistics that aid in the explanation of your answer to these questions.

16. "Fast-Food Fat Counts Full of Surprises," in USA Today, 10-20-94, compared some of the popular fast-food items in calories and fat.

Calories (x) 270 420 210 450 130 310 290 450 446 640 233

Fat (y) 9 20 10 22 6 25 7 20 20 38 11

x 552 360 838 199 360 345 552

y 55 6 20 12 36 28 22

a. Draw a scatter diagram of these data.

b. Calculate the linear coefficient, r.

c. Find the equation of the line of best fit.

d. Explain the meaning of the above answers.

x 0 3 2 2 1 3 2 4 1 1

y 25 80 45 40 36 75 50 95 30 15

a. Draw a scatter diagram of these data.

b. Calculate the correlation coefficient.

c. Find the equation of the line of best fit.

d. Explain the meaning of the above answers.

18. Investors in mutual funds keep a sharp eye on the total return on their money. They also are aware of the risk involved in their investment, commonly measured by a fund’s volatility (the greater the volatility, the higher the risk). Below is a list of 30 mutual funds randomly selected in 1998 from Fortune’s list of stock and bond funds, together with their 5-year total return (%) and risk assessment:

a. Draw a scatter diagram with 5-year total return as the y-axis and risk as the x-axis.

b. Calculate the correlation coefficient.

c. Would you conclude that in order to obtain higher total returns, mutual fund investors must take greater risks? Explain.