Section 4.1 Exercises

CLARIFYING THE CONCEPTS

Question 4.1

1. When investigating the relationship between two quantitative variables, what graph should you use first? (p. 188)

4.1.1

Scatterplot

Question 4.2

2. In your own words, explain what the correlation coefficient measures. What is the symbol that we use for the correlation coefficient? (p. 192)

Question 4.3

3. What is the range of values the correlation coefficient can take? (p. 195)

4.1.3

Between −1 and 1, inclusive

Question 4.4

4. What do the following values of indicate about the relationship between two variables? What can we say about the variables?

  1. A value of close to 1 (p. 196)
  2. A value of close to −1 (p. 197)
  3. A value of close to 0 (p. 197)

Question 4.5

5. Why do we call the predictor variable? (p. 188)

4.1.5

Often, the value of the variable can be used to predict or estimate the value of the variable.

Question 4.6

6. Suppose two quantitative variables have a positive relationship. What can we say about the values of the variable as the variable increases? (p. 190)

Question 4.7

7. Suppose two quantitative variables have a negative relationship. What can we say about the values of the variable as the variable increases? (p. 190)

4.1.7

They decrease.

Question 4.8

8. Suppose that the correlation coefficient equals 0. Does this mean that and have no relationship? Explain. (pp. 192, 197)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 9–12 Example 1 Predictor variables and
response variables
Exercises 13a–20a Example 2 Constructing a
scatterplot
Exercises 13b–20b
and 21–26		 Example 3 Describing the
relationship between
Exercises 13c–20c
and 31–34		 Example 4 Calculating the
correlation coefficient
Exercises 13d–20d
and 27–30		 Example 5 Interpreting the
correlation coefficient

201

For Exercises 9–12, two variables are given. Determine which variable is the predictor variable and which is the response variable .

Question 4.9

9. The heights (in inches) and weights (in pounds) of a sample of five women are recorded.

4.1.9

Predictor (): Height

Response (): Weight

Question 4.10

10. The number of days absent from a class, and the course grade.

Question 4.11

11. The cost of a repair job and the number of hours spent on the repair.

4.1.11

Predictor (): The number of hours spent on the repair

Response (): The cost of a repair job

Question 4.12

12. Attendance at a baseball game (in thousands), and the amount of rainfall at a baseball stadium that day (in inches).

For Exercises 13–20, do the following for the indicated data set:

  1. Construct a scatterplot of the relationship between and .
  2. Interpret the scatterplot.
  3. Calculate the correlation coefficient .
  4. Interpret the value of the correlation coefficient .

Question 4.13

13. The values of and are as follows:

10 2
20 2
30 3
40 4
50 4

4.1.13

(a)

image

(b) The variables and have a positive linear relationship. (c) (d) This value of is very close to the maximum value . We would therefore say that and are positively correlated. As increases, also tends to increase.

Question 4.14

14. The values of and are given as follows:

1 10
2 9
3 8
4 8
5 7

Question 4.15

15. The predictor variable and the response variable take the following values:

−5 −5
−5 −15
0 −20
5 −25
5 −30

4.1.15

(a)

image

(b) The variables and have a negative linear relationship. (c) (d) This value of is very close to the minimum value . We would therefore say that and are negatively correlated. As increases, tends to decrease.

Question 4.16

16. The predictor variable and the response variable take the following values:

0 10
20 10
40 15
60 20
80 20

Question 4.17

17. The heights (in inches) and weights (in pounds) of a sample of five women are recorded:

Height Weight
66 122
67 133
69 153
68 138
65 125

4.1.17

(a)

image

(b) The variables height and weight have a positive linear relationship. (c) (d) This value of is very close to the maximum value .

We would therefore say that height and weight are positively correlated. As height increases, weight also tends to increase.

Question 4.18

18. The number of days absent from a class, and the course grade:

Days absent Course grade
0 95
2 90
4 85
6 70
8 60

Question 4.19

19. The cost of a repair job, and the number of hours spent on the repair:

Cost Hours
120 2
180 3
230 5
350 8
380 10

4.1.19

(a)

image

(b) The hours a repair job takes and the cost of the repair job have a positive linear relationship. (c) (d) This value of is very close to the maximum value . We would therefore say that hours worked and cost are positively correlated. As the number of hours worked increases, the cost also tends to increase.

Question 4.20

20. Attendance at a baseball game (in thousands), and the amount of rainfall at a baseball stadium that day (in inches):

Rain Attendance
0.0 40
0.0 42
0.1 38
0.5 30
1.0 20

For Exercises 21–26, do the following:

  1. Characterize the relationship between and .
  2. State what happens to the values of the variable as the -values increase.

Question 4.21

21. image

4.1.21

(a) Strong negative linear relationship (b) They decrease.

202

Question 4.22

22. image

Question 4.23

23. image

4.1.23

(a) Moderate positive linear relationship (b) They increase.

Question 4.24

24. image

Question 4.25

25. image

4.1.25

(a) Perfect negative linear relationship (b) They decrease.

Question 4.26

26. image

For Exercises 27–30, identify which of the scatterplots in i–iv represents the data set with the following correlation coefficients:

  1. image
  2. image
  3. image
  4. image

Question 4.27

27. Near 1

4.1.27

i

Question 4.28

28. Near zero

Question 4.29

29. Near −0.5

4.1.29

iii

Question 4.30

30. Near −1

In Exercises 31–34, the values for and in each scatterplot are integer-valued. For each scatterplot, (a) reconstruct the original data set, and (b) calculate the correlation coefficient for the data.

Question 4.31

31. The data in scatterplot i

4.1.31

(a) (1,1), (2,3), (3,3), (4,4), (5,6), (6,6), (7,7), (8,7), (9,9), (10,11)

(b) Minitab: Pearson correlation of and . TI-83/84:

Question 4.32

32. The data in scatterplot ii

Question 4.33

33. The data in scatterplot iii

4.1.33

(a) (1,7), (2,8), (3,7), (4,6), (5,6), (6,5), (7,6), (8,5), (9,7), (10,6) (b) Minitab: Pearson correlation of and . TI-83/84:

Question 4.34

34. The data in scatterplot iv

203

APPLYING THE CONCEPTS

For Exercises 35–49, do the following:

  1. Construct a scatterplot of the relationship between and .
  2. Interpret the scatterplot.
  3. Calculate the correlation coefficient .
  4. Interpret the value of the correlation coefficient .

Question 4.35

videogamereg

35. Video Game Sales. The Chapter 1 Case Study looked at video game sales for the top 30 video games. The following table contains the weeks on the top 30 list and the total sales (, in millions of game units).

Video game Weeks Total sales
in millions
of units
Super Mario Bros. U
for WiiU		 78 1.7
NBA 2K14 for PS4 27 0.6
Battlefield 4 for PS3 29 0.9
Titanfall for Xbox One 10 1.2
Yoshi's New Island
for 3DS		 10 0.2
Table 4.15: Source: www.vgchartz.com.

4.1.35

(a)

image

(b) The number of weeks on the top 30 list and the total sales in millions of game units have a positive linear relationship. (c) (d) This value of is close to the maximum value . We would therefore say that weeks and total sales are positively correlated. As the number of weeks on the top 30 lists increases, the total sales also tends to increase.

Question 4.36

edunemploy

36. Does It Pay to Stay in School? The U.S. Census Bureau reported the following unemployment rates associated with the given years of education .

5.0 16.8
7.5 17.1
8.0 15.3
10.0 20.6
12.0 11.7
14.0 8.1
16.0 3.8

Question 4.37

dartsdjia

37. Darts and the Dow Jones. The following table contains a random sample of eight days from the Chapter 3 Case Study data set, indicating the stock market gain or loss for the portfolio chosen by the random darts , as well as the Dow Jones Industrial Average (DJIA) gain or loss for that day .

Darts DJIA
−27.4 −12.8
18.7 9.3
42.2 8.0
−16.3 −8.5
11.2 15.8
28.5 10.6
1.8 11.5
16.9 −5.3
Table 4.17: Source: The Wall Street Journal.

4.1.37

(a)

image

(b) The change in price of the stocks in the DJIA and the change in price of the stocks in the portfolio selected by the darts have a positive linear relationship. (c) (d) This value of is close to the maximum value . We would therefore say that DJIA and Darts are positively correlated. As the change in stock prices of the stocks in the DJIA increases, the change in stock prices selected by darts also tends to increase.

Question 4.38

ageheight

38. image Age and Height. The following table provides a random sample from the Chapter 4 Case Study data set body_females, showing the age and height of eight women.

Age Height
40 63.5
28 63.0
25 64.4
34 63.0
26 63.8
21 68.0
19 61.8
24 69.0
Table 4.18: Source: Journal of Statistics Education.

Question 4.39

gardasilreg

39. Gardasil Shots and Age. The accompanying table shows a random sample of 10 patients from the Chapter 5 Case Study data set, Gardasil, including the age of the patient and the number of shots taken by the patient .

Age Shots
13 3
21 3
16 3
17 2
17 3
18 1
25 2
15 3
12 1
16 1
Table 4.19: Source: Journal of Statistics Education.

4.1.39

(a)

image

(b) Age and number of shots have no apparent relationship. (c) (d) This value of is very close to the value . We would therefore say that there is no linear relationship between age and shots.

Question 4.40

ncaa2014

40. NCAA Power Ratings. The accompanying table shows the top 10 teams' winning percentages and power ratings for the 2013–2014 NCAA basketball season, according to www.teamrankings.com.

204

Team Winning
proportion 		Power
rating
Florida 0.923 121.2
Wichita State 0.971 119.1
Arizona 0.868 118.8
Louisville 0.838 117.9
Connecticut 0.800 117.2
Virginia 0.811 116.8
Wisconsin 0.789 116.6
Villanova 0.853 116.4
Michigan State 0.763 115.9
Michigan 0.757 115.9

Question 4.41

satfatcorr

41. Saturated Fat and Calories. The table contains the calories and saturated fat in a sample of 10 food items.

Food item Calories Grams of
saturated fat
Chocolate bar (1.45 ounces) 216 7.0
Meat & veggie pizza (large
slice)		 364 5.6
New England clam chowder
(1 cup)		 149 1.9
Baked chicken drumstick (no
skin, medium size)		 75 0.6
Curly fries, deep-fried
(4 ounces)		 276 3.2
Wheat bagel (large) 375 0.3
Chicken curry (1 cup) 146 1.6
Cake doughnut hole (one) 59 0.5
Rye bread (1 slice) 67 0.2
Raisin bran cereal (1 cup) 195 0.3
Table 4.21: Source: Food-a-Pedia.

4.1.41

(a)

image

(b) The number of calories in a serving of food and the grams of saturated fat in a serving of food have a positive linear relationship. (c) (d) This value of is positive. We would therefore say that the number of calories in a serving of food and the grams of saturated fat in a serving of food are positively correlated. As the number of calories in a serving of food increases, the grams of saturated fat in a serving of food also tends to increase.

Question 4.42

displacement

42. Engine Displacement and Gas Mileage. The table provides the engine displacement (size, in liters) and the city mpg (miles per gallon) gas mileage of a random sample of 12 vehicles taken from the Chapter 6 Case Study data set, FuelEfficiency.

Vehicle Engine
displacement		 City mpg
GMC Yukon Denali 6.2 13
Ford E350 Wagon 5.4 11
BMW435i Coupe 3.0 20
Land Rover Range Rover 5.0 13
Infiniti Q50a 3.7 19
Dodge Journey 3.6 17
Jaguar XF 5.0 15
Dodge Challenger 6.4 14
Toyota Highlander Hybrid 3.5 28
Mercedes-Benz S550 4.7 17
Ford Fiesta 1.6 29
Hyundai Elantra 2.0 24

Question 4.43

collegecompleters

43. Completing College. The twenty-first century economy not only needs students to attend college; it needs students to complete their college degrees, in order to compete in the information age. The table contains a sample of 10 states, with data on the percentage of residents who have attended college and the percentage of college attendees who have completed their college degrees .

State
California 30.9 38.8
Florida 26.6 35.5
Georgia 30.1 34.1
Illinois 35.7 39.1
Massachusetts 45.2 45.9
New York 38.7 42.8
North Carolina 30.1 35.5
Ohio 28.4 37.1
Pennsylvania 32.5 40.2
Texas 26.2 32.2
Table 4.23: Source: American Community Survey.

4.1.43

(a)

image

(b) The percent of residents of a state who attend college and the percent of residents of a state who graduate from college have a positive linear relationship. (c) (d) This value of is very close to the maximum value . We would therefore say that college attenders and college completers are positively correlated. As the number of college attenders increases, the number of college completers also tends to increase.

Question 4.44

walkbike

44. Walking or Biking to Work. In these days of high gas prices, it is worth considering alternative methods of commuting. The table contains, for a sample of 10 American cities, the percentage of people who walk to work and the percentage of people who bike to work .

City
Anaheim 1.8 0.9
Baltimore 6.5 0.8
Buffalo 6.2 0.9
Cincinnati 5.4 0.5
Detroit 3.1 0.3
Jacksonville 1.4 0.4
Las Vegas 1.9 0.4
New Orleans 5.1 2.1
Orlando 1.9 0.4
Sacramento 3.2 2.5
Table 4.24: Source: U.S. Census Bureau.

Question 4.45

teenbirth

45. Teenage Birth Rate. The National Center for Health Statistics publishes data on state birth rates. The table contains the overall birth rate and the teenage birth rate for eight randomly chosen states. The overall birth rate is defined by the NCHS as “live births per 1000 women,” and the teenage birth rate is defined as “live births per 1000 women aged 15–19.”

205

State
California 62.0 23.6
Florida 59.3 24.6
Georgia 61.6 30.5
New York 58.8 17.7
Ohio 62.7 27.2
Pennsylvania 58.4 20.9
Texas 69.9 41.0
Virginia 60.9 20.1

4.1.45

(a)

image

(b) The overall birth rate and the teenage birth rate have a positive linear relationship. (c) (d) This value of is very close to the maximum value . We would therefore say that overall birth rate and teen birth rate are positively correlated. As the overall birth rate increases, the teen birth rate also tends to increase.

Question 4.46

brainbody

46. Brain and Body Weight. A study compared the body weight (in kilograms) and brain weight (in grams) for a sample of mammals, with the results shown in the following table.2

52.16 440.0
60.00 81.0
27.66 115.0
85.00 325.0
36.33 119.5
100.00 157.0
35.00 56.0
62.00 132.0
83.00 98.2
55.50 175.0

Question 4.47

consumersentiment

47. Consumer Sentiment. Would you expect the consumer sentiment (a measure of how upbeat a consumer feels about his or her personal economic condition) of those with lower incomes to be correlated with that of those with higher incomes, over time? The University of Michigan's Survey of Consumers published the data in the following table, showing the consumer sentiment in 2013 month by month for the two groups.

Month
Jan 71.6 80.2
Feb 75.7 82.4
Mar 78.3 83.7
Apr 74.5 79.8
May 80.3 94.1
Jun 76.1 98.9
Jul 82.4 90.0
Aug 78.0 89.6
Sep 72.3 86.2
Oct 71.4 77.0
Nov 67.9 88.7
Dec 78.9 88.0

4.1.47

(a)

image

(b) The consumer sentiment for incomes under $75,000 and the consumer sentiment for incomes $75,000 or higher have a positive linear relationship. (c) (d) This value of is positive. We would therefore say that the consumer sentiment for incomes under $75,000 and the consumer sentiment for incomes $75,000 or higher are positively correlated. As the consumer sentiment for incomes under $75,000 increases, the consumer sentiment for incomes $75,000 or higher also tends to increase.

Question 4.48

satlanguages

48. SAT Scores, by Foreign Language. The table contains the mean 2014 SAT Critical Reading and Math scores, which are categorized by the foreign language taken in high school or spoken at home.

Language SAT Critical
Reading score		 SAT Mathematics
score
Chinese 535 606
French 519 525
German 530 540
Greek 526 543
Hebrew 526 541
Italian 497 509
Japanese 521 552
Korean 490 576
Latin 556 556
Russian 483 535
Spanish 498 508

Question 4.49

batters2014

49. Batting Average and Runs Scored. The table shows the top 10 hitters in the American League of Major League Baseball for 2014. We are interested in estimating the number of runs scored using the player's batting average .

Batter Team Runs
scored		 Batting
average
Jose Altuve Houston Astros 85 0.341
Victor Martinez Detroit Tigers 87 0.335
Michael Brantley Cleveland
Indians		 94 0.327
Adrian Beltre Texas Rangers 79 0.324
Jose Abreu Chicago White
Sox		 80 0.317
Robinson Cano Seattle Mariners 77 0.314
Miguel Cabrera Detroit Tigers 101 0.313
Melky Cabrera Toronto Blue
Jays		 81 0.301
Adam Eaton Chicago White
Sox		 76 0.300
Howie Kendrick Los Angeles
Angels		 85 0.293

4.1.49

(a)

image

(b) The batting average of a baseball player and the number of runs scored by a baseball player have no apparent relationship. (c) (d) This value of is very close to the value . We would therefore say that there is no linear relationship between batting average and runs scored.

Correlation in Accounting. A company's current ratio measures its ability to pay its short-term obligations. Use the data in the table, which contains a random sample of large technology companies in 2010, for Exercises 50–56. Total assets and total liabilities are in billions of dollars.

206

Company Current
ratio		 Price-
earnings ratio		 Assets Liabilities
Microsoft 1.82 12.51 77.9 38.3
Intel 2.79 18.44 53.1 11.4
Dell 1.28 10.95 33.7 28.0
Apple 1.88 24.57 53.9 26.0
Google 10.62 18.87 40.5 4.5
Table 4.30: Source: Lexis Nexis.

Question 4.50

accountingcorr

50. Provide and interpret a scatterplot of liabilities versus assets.

Question 4.51

accountingcorr

51. Calculate and interpret the correlation coefficient between liabilities and assets.

4.1.51

; This value of is positive. We would therefore say that assets and liabilities are positively correlated. As assets increase, liabilities also tend to increase.

Question 4.52

accountingcorr

52. Provide and interpret a scatterplot of current ratio versus price-earnings ratio.

Question 4.53

accountingcorr

53. Calculate and interpret the correlation coefficient between current ratio and price-earnings ratio.

4.1.53

; This value of is very close to the value . We would therefore say that there is no linear relationship between price–earnings ratio and current ratio.

Question 4.54

accountingcorr

54. Compute a new variable, called net worth, which equals assets – liabilities.

Question 4.55

accountingcorr

55. Provide and interpret a scatterplot of net worth versus current ratio.

4.1.55

image

The current ratio of a company and the company's net worth have a positive linear relationship.

Question 4.56

accountingcorr

56. Calculate and interpret the correlation coefficient between net worth and current ratio.

Best Places for Dating. Sperling's Best Places published the list of best places for dating in America. The table shows the top 10 places, along with the overall dating score and a set of predictor variables. Use this information for Exercises 57–59.

City Percentage
18–24 years
old		 Percentage
18–24 who
are single		 Online
dating
score
Austin 100.0 13.4% 81.2% 77.8
Colorado
Springs		 88.7 10.5% 74.2% 88.9
San Diego 84.0 11.3% 79.4% 77.4
Raleigh 80.7 11.6% 82.9% 79.2
Seattle 78.7 9.0% 83.9% 100.0
Charleston 78.7 11.2% 82.7% 66.9
Norfolk 77.0 11.2% 75.6% 82.9
Ann Arbor 75.5 12.9% 90.3% 51.1
Springfield 75.2 11.7% 89.8% 63.5
Honolulu 75.2 10.1% 82.3% 50.2

Question 4.57

bestdating

57. Construct and interpret a scatterplot for each of the following predictor variables versus the overall dating score:

  1. Percentage 18–24 years old
  2. Percentage 18–24 years old who are single
  3. Online dating score

4.1.57

(a)

image

The percentage of 18- to 24-year-olds in a city and the overall dating score in a city have a positive linear relationship.

(b)

image

The percentage of a city's residents who are 18- to 24-year-olds and single and the overall dating score of a city have a negative linear relationship.

(c)

image

The online dating score of a city and the overall dating score of a city have a positive linear relationship.

Question 4.58

bestdating

58. Calculate and interpret the correlation coefficient between each of the following predictor variables and the overall dating score:

  1. Percentage 18–24 years old
  2. Percentage 18–24 years old who are single
  3. Online dating score

Question 4.59

bestdating

59. Based on your work in Exercises 57 and 58, which predictor variable is the best indicator of the overall dating score, and thus an indicator of the best places for dating?

4.1.59

Percentage of 18- to 24-year-olds who are single

Virginia Weather. The table contains data on weather in a sample of cities in the state of Virginia. Use this information for Exercises 60–65.

Data on the weather in Virginia
City Average
January
temperature		 Heating
degree-
days		 Average
July
temperature		 Cooling
degree-
days
Alexandria 34.9 4055 79.2 1531
Arlington 34.9 4055 79.2 1531
Blacksburg 30.9 5559 71.1 533
Charlottesville 35.5 4103 76.9 1212
Chesapeake 40.1 3368 79.1 1612
Danville 36.6 3970 78.8 1418
Hampton 39.4 3535 78.5 1432
Harrisonburg 30.5 5333 73.5 758
Leesburg 31.5 5031 75.2 911
Lynchburg 34.5 4354 75.1 1075
Manassas 31.7 4925 75.7 1075
Newport News 41.2 3179 80.3 1682
Norfolk 40.1 3368 79.1 1612
Petersburg 39.7 3334 79.6 1619
Portsmouth 40.1 3368 79.1 1612
Richmond 36.4 3919 77.9 1435
Roanoke 35.8 4284 76.2 1134
Suffolk 39.6 3467 78.5 1427
Virginia Beach 40.7 3336 78.8 1482
Table 4.32: Source: National Oceanic and Atmospheric Administration.

Question 4.60

vaweather

60. Construct and interpret a scatterplot for the average January temperature versus the cooling degree-days.

Question 4.61

vaweather

61. Based on your scatterplot, what would you expect the sign of the correlation coefficient to be? Why?

4.1.61

Positive. The scatterplot indicates a positive linear relationship between cooling-degree days and the average January temperature.

Question 4.62

vaweather

62. Calculate the correlation coefficient between the average January temperature and the cooling degree-days.

Question 4.63

vaweather

63. Build and interpret a scatterplot for the average July temperature versus the heating degree-days.

4.1.63

image

Heating degree-days and the average July temperature have a negative linear relationship.

Question 4.64

vaweather

64. Calculate the correlation coefficient between the average July temperature and the heating degree-days.

Question 4.65

vaweather

65. Which relationship would you say is stronger: the relationship between the average January temperatures with the cooling degree-days, or the relationship between the average July temperatures and the heating degree-days?

4.1.65

The relationship between the average July temperatures and the heating degree-days is stronger.

What's Your Major? The table contains the percentages of students majoring in (i) Mathematics/Statistics/Computer Science, (ii) Biological Sciences, and (iii) Psychology, for a sample of 10 states. Use this information for Exercises 66–71.

207

State Math/Stat/
Comp Sci		 Biological
Sci		 Psychology
Alaska 2.0 11.0 5.5
Connecticut 4.1 5.3 5.7
Hawaii 2.9 6.9 5.2
Idaho 3.7 9.6 3.2
Maryland 5.7 7.1 4.8
Montana 2.7 11.1 4.1
New Jersey 5.3 5.3 4.7
Oregon 3.2 8.4 5.4
Virginia 5.5 5.4 4.7
Wyoming 2.2 12.2 3.6
Table 4.33: Source: American Community Survey.

Question 4.66

statemajors

66. Construct a scatterplot of the Math/Stat/Comp Sci majors versus the Psychology majors.

Question 4.67

statemajors

67. Based on your scatterplot, would you say that a relationship exists between the two sets of majors?

4.1.67

No.

Question 4.68

statemajors

68. Calculate the correlation coefficient between the Math/Stat/Comp Sci majors and the Psychology majors. Was your intuition in Exercise 67 confirmed?

Question 4.69

statemajors

69. Construct a scatterplot of the Biological Science majors versus the Psychology majors.

4.1.69

image

Question 4.70

statemajors

70. Based on your scatterplot in Exercise 69, would you say that a relationship exists between the two sets of majors?

Question 4.71

statemajors

71. Calculate the correlation coefficient between the Biological Science majors and the Psychology majors. Was your intuition in Exercise 70 confirmed?

4.1.71

. Yes.

Worldwide Indicators of Well-Being. The Statistics Online Computational Resource (SOCR) provides resources for statistics students and educators. The following table, sampled from data provided by SOCR, includes 10 countries, along with 3 indicators of their well-being: an indicator of economic dynamism, a measure of literacy, and a health indicator. Use this information for Exercises 72–82.

Country Economy Literacy Health
Australia 71.5 91.5 95.2
Canada 68.5 96.7 92.8
Germany 61.9 91.1 92.8
India 50.0 66.2 51.7
Japan 69.0 94.0 100.0
Mexico 42.0 74.7 78.3
Pakistan 41.5 66.9 49.3
South Korea 73.0 96.7 87.9
United Kingdom 72.9 92.8 90.3
United States 77.8 89.4 85.5

Question 4.72

wellbeing

72. Some analysts claim that literacy depends on the economy of a country. Provide a scatterplot of literacy versus economy . Describe any relationship between the variables.

Question 4.73

wellbeing

73. Calculate and interpret the correlation coefficient for the linear relationship between and .

4.1.73

. This value of is very close to the maximum value . We would therefore say that economy and literacy are positively correlated. As the economy increases, literacy also tends to increase.

Question 4.74

wellbeing

74. Other analysts state that a country's economy depends on the country's literacy. Build a scatterplot of economy versus literacy . Describe any relationship between the variables. Compare this scatterplot with the one in Exercise 72. Discuss similarities and differences.

Question 4.75

wellbeing

75. Do you expect that the correlation coefficient between and will be the same as that between and ? Calculate and interpret the correlation coefficient for the linear relationship between and . Was your intuition confirmed?

4.1.75

Yes. . Yes.

Question 4.76

wellbeing

76. Provide a scatterplot of literacy versus health . Describe any relationship between the variables.

Question 4.77

wellbeing

77. Calculate and interpret the correlation coefficient for the linear relationship between and .

4.1.77

. This value of is very close to the maximum value . We would therefore say that health and literacy are positively correlated. As health increases, literacy also tends to increase.

Question 4.78

wellbeing

78. Without calculating it, state what the correlation coefficient would be for the linear relationship between and .

Question 4.79

wellbeing

79. Provide a scatterplot of health versus economy . Describe any relationship between the variables.

4.1.79

image

Economy and health have a positive linear relationship.

Question 4.80

wellbeing

80. Calculate and interpret the correlation coefficient for the linear relationship between and .

Question 4.81

wellbeing

81. Without calculating it, state what the correlation coefficient would be for the linear relationship between and .

4.1.81

Question 4.82

wellbeing

82. Based on your work in the previous exercises, state a general rule about the correlation between and , and the correlation between and .

Question 4.83

83. Computational Formula for . The following computational formula may be used as an equivalent of the definition formula for the correlation coefficient :

Use the computational formula and the TI-83/84 to calculate the correlation coefficient for the relationship between square footage and sales price of the eight home lots for sale in Glen Ellyn from Example 2 (page 189).

4.1.83

image sqrfootsale

BRINGING IT ALL TOGETHER

Lead and Zinc Concentrations in River Fish. Like to go fishing? In some areas, it may not be healthy to eat the fish you catch, due to the pollutants in the river that the fish ingest. Use the information in the table for Exercises 84–88. The table contains the lead and zinc concentrations in river fish from the Spokane River in Washington State, in parts per million.3

208

Fish

1 0.73 45.3
2 1.14 50.8
3 0.60 40.2
4 1.59 64.0
5 4.34 150.0
6 1.98 106.0
7 3.12 90.8
8 1.80 58.8
9 0.65 35.4
10 0.56 28.4

Question 4.84

leadzincfish

84. Investigate the relationship.

  1. Construct a scatterplot of the variables. Make sure the variable goes on the y axis.
  2. What type of relationship do these variables have: positive, negative, or no apparent linear relationship?
  3. Will the correlation coefficient be positive, negative, or near zero?

Question 4.85

leadzincfish

85. Calculate and interpret the correlation coefficient.

  1. Compute the value of the correlation coefficient.
  2. Does this value for concur with your judgment in part (c) of the previous exercise?
  3. Interpret the meaning of this value of the correlation coefficient.

4.1.85

(a) (b) Yes. (c) This value of is very close to the maximum value . We would therefore say that lead and zinc are positively correlated. As the lead content in the fish increases, the zinc content in the fish also tends to increase.

Question 4.86

leadzincfish

86. Determine whether we can conclude that and are correlated.

Question 4.87

leadzincfish

87. Transformation. Add 5 to each value for .

  1. Redraw the scatterplot. Comment on the similarity or difference from the scatterplot in Exercise 84(a).
  2. Recalculate the correlation coefficient.
  3. Compare your answers from Exercises 85(a) and 87(b).
  4. Compose a rule that states the behavior of the correlation coefficient when a constant is added to each data value.

4.1.87

(a)

image

Everything is the same except the dots are shifted up 5 ppm. (b) (c) They are the same. (d) When a constant is added to each -data value the correlation coefficient stays the same.

Question 4.88

leadzincfish

image 88. Transformation. What if, starting with the original data in the table, we added a certain unknown constant amount to each value for ?

  1. Without redrawing the scatterplot, describe how this change would affect the scatterplot you drew in Exercise 84(a).
  2. Without recalculating the correlation coefficient, state what you think the effect of this change would be on the correlation coefficient. Why do you think that?
  3. Compose a rule that states the behavior of the correlation coefficient when a constant is added to each data value.

CONSTRUCT YOUR OWN DATA SETS

Question 4.89

89. Describe two variables from real life that would have a value of close to 1. Explain why they are positively correlated.

4.1.89

Answers will vary.

Question 4.90

90. Create a sample of five observations from each of your variables in the previous exercise, and put them into a table similar to Table 4 (page 194). Next, construct a scatterplot of the variables. Finally, draw a single straight line through the data points in the plot in a manner that you think best approximates the relationship between the variables.

Use the Correlation and Regression applet for Exercises 91–93.

Question 4.91

91. Create a set of points such that the correlation coefficient takes approximately the following values.

Note that you can drag points up or down to adjust your value of .

4.1.91

(a)–(c) Answers will vary.

Question 4.92

92. Describe the relationship between the variables for each of the sets of points in the previous exercise.

Question 4.93

93. Select “Show mean and mean lines.” Create a set of points such that the correlation coefficient takes approximately the following values. Note that you can drag points up or down to adjust your value of .

4.1.93

(a)–(c) Answers will vary.

WORKING WITH LARGE DATA SETS

image Chapter 4 Case Study: Measuring the Human Body. Open the data sets body_females body_males. We shall explore the relationships between height and weight for the women and men in these data sets using the tools and techniques we have learned in this section. Use technology to do the following:

Question 4.94

body_females

body_males

94. Construct a scatterplot of weight versus height for the men.

Question 4.95

body_females

body_males

95. Characterize the relationship between male height and weight.

Question 4.96

body_females

body_males

96. Find the correlation coefficient between height and weight for the males.

Question 4.97

body_females

body_males

97. Interpret the correlation coefficient for men's height and weight.

Question 4.98

body_females

body_males

98. Discuss similarities and differences among your results for the females (from the Your Turn exercises in this section) and the males, as follows:

  1. The scatterplots
  2. The relationships between height and weight
  3. The correlation coefficients