13 Inference in Regression

Section 13.1 Exercises

CLARIFYING THE CONCEPTS

Question 13.1

1. What is the difference between the regression equation (calculated using the sample) and the population regression equation? (p. 718)

13.1.1

The regression equation is calculated from a sample and is valid only for values of in the range of the sample data. The population regression equation may be used to approximate the relationship between the predictor variable and the response variable for the entire population of (, ) pairs.

Question 13.2

2. What are the four regression model assumptions? (p. 718)

Question 13.3

3. How do we go about verifying the regression model assumptions? (p. 719)

13.1.3

We construct a scatterplot of the residuals against the fitted values and a normal probability plot of the residuals. We must make sure that the scatterplot contains no strong evidence of any unhealthy patterns and that the normal probability plot indicates no evidence of departures from normality in residuals.

Question 13.4

4. What is the difference between and on the one hand and and on the other hand? (p. 718)

Question 13.5

5. What does it mean for the relationship between and when equals 0? (p. 721)

13.1.5

There is no relationship between and .

Question 13.6

6. What is the difference between and ? (p. 722)

PRACTICING THE TECHNIQUES

CHECK IT OUT!

To do	Check out	Topic
Exercises 7–14	Example 2	Calculating the residuals and verifying the regression assumptions
Exercises 15–18	Example 4	Test for the slope : critical-value method
Exercises 19–22	Example 5	Test for the slope : -value method
Exercises 23–30	Example 6	Confidence interval for the slope
Exercises 31–38	Example 7	Using confidence intervals to perform the test for the slope

For Exercises 7–14, you are given the regression equation.

Calculate the predicted values.
Compute the residuals.
Construct a scatterplot of the residuals versus the predicted values.
Use technology to construct a normal probability plot of the residuals.
Verify that the regression assumptions are valid.

Question 13.7


1	15
2	20
3	20
4	25
5	25

13.1.7

(a) and (b)

		Predicted value	Residual ()
1	15	16	–1
2	20	18.5	1.5
3	20	21	–1
4	25	23.5	1.5
5	25	26	–1

(c) and (d) See Solutions Manual. (e) The scatterplot of the residuals contains an unhealthy pattern, so the regression assumptions are not verified.

Question 13.8


0	10
5	20
10	45
15	50
20	75

732

Question 13.9


−5	0
−4	8
−3	8
−2	16
−1	16

13.1.9

(a) and (b)

		Predicted value	Residual ()
–5	0	1.6	–1.6
–4	8	5.6	2.4
–3	8	9.6	–1.6
–2	16	13.6	2.4
–1	16	17.6	–1.6

(c) and (d) See Solutions Manual. (e) The scatterplot of the residuals contains an unhealthy pattern, so the regression assumptions are not verified.

Question 13.10

10.


−3	−5
−1	−15
1	−20
3	−25
5	−30

Question 13.11

11.


10	100
20	95
30	85
40	85
50	80

13.1.11

(a) and (b)

		Predicted value	Residual ()
10	100	99	1
20	95	94	1
30	85	89	–4
40	85	84	1
50	80	79	1

(c) and (d) See Solutions Manual. (e) The scatterplot of the residuals contains an unhealthy pattern, so the regression assumptions are not verified.

Question 13.12

12.


0	11
20	11
40	16
60	21
80	26

Question 13.13

13.


1	1
2	1
3	2
4	3
5	3

13.1.13

(a) and (b)


1	1	0.8	0.2
2	1	1.4	–0.4
3	2	2	0
4	3	2.6	0.4
5	3	3.2	–0.2

(c)

(d)

(e) The scatterplot in (c) of the residuals versus fitted values shows no strong evidence of unhealthy patterns. Thus, the independence assumption, the constant variance assumption, and the zero-mean assumption are verified. Also, the normal probability plot of the residuals in (d) indicates no evidence of departure from normality of the residuals. Therefore we conclude that the regression assumptions are verified.

Question 13.14

14.


1	6
2	5
2	4
2	3
3	2

For Exercises 15–18, follow these steps. Assume that the regression model assumptions are valid.

Find for a two-tailed test with level of significance and .
Calculate .
Compute .
Calculate .
Perform the hypothesis test for the linear relationship between and , using the critical-value method and .

Question 13.15

15. Data in Exercise 7, where

13.1.15

(a) (b) (c) (d) (e) : There is no linear relationship between and . : There is a linear relationship between and . Reject if or . Since , we reject . There is evidence at level of significance that and that there is a linear relationship between and .

Question 13.16

16. Data in Exercise 8, where

Question 13.17

17. Data in Exercise 9, where

13.1.17

Question 13.18

18. Data in Exercise 10, where

For Exercises 19–22, follow these steps. Assume that the regression model assumptions are valid.

Calculate .
Compute .
Calculate .
Find .
Perform the hypothesis test for the linear relationship between and using the -value method and level of significance .

Question 13.19

19. Data in Exercise 11, where

13.1.19

(a) (b) (c) (d) (e) : There is no linear relationship between and . : There is a linear relationship between and . Reject if -value . Since -value , we reject . There is evidence at level of significance that and that there is a linear relationship between and .

Question 13.20

20. Data in Exercise 12, where

Question 13.21

21. Data in Exercise 13, where

13.1.21

(a) (b) (c)

(d) (e) : There is no linear relationship between and . : There is a linear relationship between and . Reject if -value . Since -value , we reject . There is evidence at level of significance that and that there is a linear relationship between and .

Question 13.22

22. Data in Exercise 14, where

For Exercises 23–30, follow these steps. Assume that the regression model assumptions are valid.

Find for a 95% confidence interval for .
Find the margin of error .
Construct a 95% confidence interval for .

Question 13.23

23. Data in Exercise 7

13.1.23

(a) (b) (c) (0.909, 4.091)

Question 13.24

24. Data in Exercise 8

Question 13.25

25. Data in Exercise 9

13.1.25

(a) (b) (c) (1.4544, 6.5456)

Question 13.26

26. Data in Exercise 10

Question 13.27

27. Data in Exercise 11

13.1.27

(a) (b) (c) (–0.7598, −0.2402)

Question 13.28

28. Data in Exercise 12

Question 13.29

29. Data in Exercise 13

13.1.29

(a) (b) (c) (0.2326, 0.9674). TI-83/84: (0.2325, 0.9675)

Question 13.30

30. Data in Exercise 14

For Exercises 31–38, using the confidence interval from the indicated exercise, perform the test for at level of significance .

Question 13.31

31. Exercise 23

13.1.31

: There is no linear relationship between and . : There is a linear relationship between and . Since the confidence interval from Exercise 23 (c) does not contain zero, we may conclude that and that a linear relationship exists between and , at level of significance .

Question 13.32

32. Exercise 24

Question 13.33

33. Exercise 25

13.1.33

: There is no linear relationship between and . : There is a linear relationship between and . Since the confidence interval from Exercise 25 (c) does not contain zero, we may conclude that and that a linear relationship exists between and , at level of significance .

Question 13.34

34. Exercise 26

Question 13.35

35. Exercise 27

13.1.35

: There is no linear relationship between and . There is a linear relationship between and . Since the confidence interval from Exercise 27 (c) does not contain zero, we may conclude that and that a linear relationship exists between and , at level of significance .

Question 13.36

36. Exercise 28

Question 13.37

37. Exercise 29

13.1.37

: There is no linear relationship between and . There is a linear relationship between and . Since the confidence interval from Exercise 29 (c) does not contain zero, we may conclude that and that a linear relationship exists between and , at level of significance .

Question 13.38

38. Exercise 30

APPLYING THE CONCEPTS

For Exercises 39–46, assume the regression requirements are met. Test for the linear relationship between and , using level of significance .

733

Question 13.39

volweight

39. Volume and Weight. The following table contains the volume (, in cubic meters) and weight (, in kilograms) of five randomly chosen packages shipped to a local college.

Volume	Weight
4	10
8	16
12	25
16	30
20	35

13.1.39

: There is no relationship between volume () and weight (). : There is a linear relationship between volume () and weight (). Reject if the -value ≤ 0.05. Since the -value is , we reject . There is evidence for a linear relationship between volume () and weight ().

Question 13.40

familypet

40. Family Size and Pets. The number of family members in a random sample taken from a suburban neighborhood, along with the number of pets belonging to each family are shown in the following table.

Family size	Pets
2	1
3	2
4	2
5	3
6	3

Question 13.41

worldtemp

41. World Temperatures. Listed in the following table are the low and high temperatures for a particular day, measured in degrees Fahrenheit, for a random sample of cities worldwide.

City	Low	High
Kolkata	57	77
London	36	45
Montreal	7	21
Rome	39	55
San Juan	70	83
Shanghai	34	45

13.1.41

: There is no relationship between Low () and High ().

: There is a linear relationship between Low () and High ().

Reject if . Since is , we reject . There is evidence for a linear relationship between Low () and High ().

Question 13.42

videogamereg

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

Video game	Weeks	Total sales
Super Mario Bros. U for WiiU	78	1,690,689
NBA 2K14 for PS4	27	608,899
Battlefield 4 for PS3	29	911,687
Titanfall for Xbox One	10	1,150,856
Yoshi's New Island for 3DS	10	172,680

Question 13.43

dartsdjia

43. 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 gain or loss for that day .

Darts	DJIA
−27.4	−12.8
18.7	9.3
42.2	8
−16.3	−8.5
11.2	15.8
28.5	10.6
1.8	11.5
16.9	−5.3

13.1.43

: No linear relationship exists between DJIA and darts.

: A linear relationship exists between DJIA and darts.

Reject if the -value . . -value .

The is not ≤ , so we do not reject . Insufficient evidence exists, at level of significance , for a linear relationship between DJIA and darts.

Question 13.44

ageheight

44. 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 the eight women.

Age	Height
40	63.5
28	63
25	64.4
34	63
26	63.8
21	68
19	61.8
24	69

Question 13.45

gardasilreg

45. 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

13.1.45

: No linear relationship exists between age and shots.

: A linear relationship exists between age and shots.

Reject if the -value . . .

The is not , so we do not reject . insufficient evidence exists, at level of significance , for a linear relationship between a patient's age and the number of shots taken by the patient.

Question 13.46

ncaa2014

46. NCAA Power Ratings. The accompanying table shows the top 10 teams' winning percentage and power rating for the 2013-2014 NCAA basketball season, according to www.teamrankings.com.

734

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

For Exercises 47–54, do the following for the indicated data:

Calculate the margin of error for a 95% confidence interval for .
Construct a 95% confidence interval for .
Interpret the confidence interval.

Question 13.47

47. Volume and Weight. Data from Exercise 39.

13.1.47

(a) (b) (1.2688, 1.9312) (c) We are 95% confident that the interval (1.2688, 1.9312) captures the population slope of the relationship between volume and weight.

Question 13.48

48. Family Size and Pets. Data from Exercise 40

Question 13.49

49. World Temperatures. Data from Exercise 41

13.1.49

(a) (b) (0.8064, 1.2868). TI-83/84: (0.8063, 1.2868) (c) We are 95% confident that the interval (0.8063, 1.2868) captures the population slope of the relationship between Low and High.

Question 13.50

50. Video Game Sales. Data from Exercise 42

Question 13.51

51. Darts and the Dow Jones. Data from Exercise 43

13.1.51

(a) 1.5910 (b) (–0.1769, 3.0051) (c) We are 95% confident that the interval (–0.1769, 3.0051) captures the slope of the population regression line. That is, we are 95% confident that for each additional $1 the stocks in the DJIA gain in one day, the daily change in the stocks in the portfolio predicted by the darts lies between –$0.039 and $0.66248.

Question 13.52

52. Age and Height. Data from Exercise 44

Question 13.53

53. Gardasil Shots and Age. Data from Exercise 45

13.1.53

(a) 0.19824 shot (b) (–0.1826, 0.21388) (c) We are 95% confident that the interval (–0.1826, 0.21388) captures the slope of the population regression line. That is, we are 95% confident that for each additional year in a patient's age, the change in the number of shots taken by the patient lies between −0.1826 shot and 0.21388 shot.

Question 13.54

54. NCAA Power Ratings. Data from Exercise 46

Question 13.55

satfatreg

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

Construct a 90% confidence interval for the slope of the regression line, for the regression of calories on saturated fat.
Using your confidence interval, conclude whether a linear relationship exists between calories and saturated fat, with level of significance .

Food item	Calories	Grams of saturated fat
Chocolate bar (1.45 oz)	215.66	6.9618
Meat & veggie pizza, big slice (1/8 lg pizza)	363.81	5.6472
New England clam chowder (cup)	148.80	1.8600
Baked chicken drumstick (no skin, medium size)	75.24	0.6424
Curly fries, deep-fried (4 ounces)	276.21	3.1752
Wheat bagel (large)	374.66	0.2751
Chicken curry (cup)	146.32	1.5930
Cake doughnut hole (one)	58.94	0.5068
Rye bread (slice)	67.34	0.1638
Raisin Bran cereal (cup)	194.59	0.3355

Table 13.28: Source: Food-A-Pedia.

13.1.55

(a) (–6.647, 49.817) (b) 0 lies in the interval, so we do not reject .

Question 13.56

displacement

56. 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.

Construct a 95% confidence interval for the slope of the regression line, for the regression of city MPG on engine displacement.
Using your confidence interval, conclude whether a linear relationship exists between city MPG and engine displacement, with level of significance .

Vehicle	Engine displacement	City MPG
GMC Yukon Denali	6.2	13
Ford E350 Wagon	5.4	11
BMW 435i 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 S 550	4.7	17
Ford Fiesta	1.6	29
Hyundai Elantra	2.0	24

Batting Average and Runs Scored. The table shows the top 10 hitters in Major League Baseball for 2013. We are interested in estimating the number of runs scored by using the player's batting average . Use this information for Exercises 57–60.

Batter	Team	Runs scored	Batting average
Miguel Cabrera	Tigers	103	0.348
Mike Trout	Angels	109	0.323
Matt Carpenter	Cardinals	126	0.318
Andrew McCutchen	Pirates	97	0.317
Paul Goldschmidt	Diamondbacks	103	0.302
Josh Donaldson	Athletics	89	0.301
Chris Davis	Orioles	103	0.286
Carlos Gomez	Brewers	80	0.284
Manny Machado	Orioles	88	0.283
Evan Longoria	Rays	91	0.269

Question 13.57

batters2013

57. Assess the regression assumptions. Is it okay to proceed with the regression?

13.1.57

The scatterplot of the residuals versus the fitted values shows no evidence of the unhealthy patterns shown in Figure 4. Thus, the independence assumption, the constant variance assumption, and the zero-mean assumption are verified.

Also, the normal probability plot of the residuals indicates no evidence of departures from normality in the residuals. Therefore, we conclude that the regression assumptions are verified and it is okay to proceed with the regression.

Question 13.58

batters2013

58. Regress runs scored on batting average. Test for the significance of the linear relationship, using level of significance .

Question 13.59

batters2013

59. Find the residual for Matt Carpenter. What is unusual about Matt Carpenter in this regression?

13.1.59

22.51 runs; by far the highest number of runs scored but the third highest batting average.

735

Question 13.60

batters2013

60. Test for the significance of the linear relationship, using level of significance . Compare your conclusion with the earlier regression, using level of significance . How do you suggest we resolve this dilemma?

BRINGING IT ALL TOGETHER

SAT Reading and Math Scores. Use this information for Exercises 61–65. The table shows the SAT scores for five states as reported by the College Board. We are interested in whether a linear relationship exists between the SAT Reading score and the SAT Math score .

State	SAT Reading	SAT Math
New York	497	510
Connecticut	515	515
Massachusetts	518	523
New Jersey	501	514
New Hampshire	522	521

Question 13.61

statesat

61. What Result Might We Expect? Consider the accompanying scatterplot of Math score versus Reading score. Is there evidence for or against the null hypothesis that no linear relationship exists? Explain.

13.1.61

Against; the points appear to lie near a line with a positive slope.

Question 13.62

statesat

62. Consider the following graphics. Is there strong evidence that the regression assumptions are violated?

Plot of residuals versus fitted values.

Normality plot of residuals.

Question 13.63

statesat

63. Test, using level of significance , whether a linear relationship exists between the SAT reading score and the SAT Math score.

13.1.63

: There is no linear relationship between SAT Reading score () and SAT Math score (). : There is a linear relationship between SAT Reading score () and SAT Math score (). Reject if or . Since , we reject . There is evidence at level of significance that and that there is a linear relationship between SAT Reading score () and SAT Math score ().

Question 13.64

statesat

64. Construct and interpret a 90% confidence interval for a slope .

Question 13.65

statesat

65. Do your inferences in Exercises 63 and 64 agree with each other? Explain.

13.1.65

Yes. Since the confidence interval from Exercise 64 does not contain zero, we may conclude that and that a linear relationship exists between SAT Reading score () and SAT Math score (), at level of significance .

Question 13.66

66. Challenge Exercise. Suppose we have a regression equation whose slope was not significant (that is, the null hypothesis was not rejected). What if we add five new data values to the original data set, and all five data values are identical, ? How and why will this affect the following statistics? Will the statistic increase, decrease, or remain unchanged, or is there insufficient information to determine? (Hint: The data point always lies on the estimated regression line.)

Question 13.67

67. Challenge Exercise. Refer to Exercise 66. How and why will the change affect the following items?

conclusion

13.1.67

(a) increases if is positive and decreases if is negative. (b) remains the same. (c) decreases. (d) -value decreases. (e) Since we don't know what the new -value will be, we don't know if the -value will decrease enough to change the conclusion from “Do not reject ” to “Reject .”

Question 13.68

68. Challenge Exercise. Suppose a regression analysis of on was found to be significant (that is, the null hypothesis was rejected). What if we get 10 new data values, all with different values of and all of which can be found on the estimated regression line of the original model? How and why will this change affect the following statistics? Will the statistic increase, decrease, or remain unchanged, or is there insufficient information to determine?

Question 13.69

69. Refer to Exercise 68. How and why will this change affect the following measures?

conclusion

13.1.69

(a) increases if is positive and decreases if is negative.

(b) increases. (c) decreases. (d) -value decreases. (e) Unchanged.

Question 13.70

70. Challenge Exercise. Suppose a regression analysis of on was found to be significant (that is, the null hypothesis was rejected) and the slope . Consider the observation , which represents the data value for the maximum value of in the data set. Suppose the residual for is negative. What if we increase max by an arbitrary amount so that the new data value is ? (All other data values in the data set are unchanged.) How will this increase affect the following measures? Will they increase, decrease, or remain unchanged, or is there insufficient information to determine the effect?

736

Question 13.71

71. Challenge Exercise. Refer to Exercise 70. How and why will the change affect the following measures?

conclusion

13.1.71

(a-b) Decrease (c-d) Increase (e) Depends on the new -value.

WORKING WITH LARGE DATA SETS

For Exercises 72–74, use technology to solve the following problems:

Verify the regression model assumptions.
Construct and interpret a 95% confidence interval for .
Based on the confidence interval constructed in (b), would you expect the hypothesis test to reject the null hypothesis that ?
Test, at , whether a linear relationship exists between and .

Question 13.72

darts

72. Open the Darts data set, which we used for the Chapter 3 Case Study. Use the Dow Jones Industrial Average to estimate the pros' performance .

Question 13.73

nutrition

73. Open the Nutrition data set. Estimate the number of calories per gram using the amount of fat per gram .

13.1.73

(a) The scatterplot of the residuals contains evidence of an unhealthy pattern and the normal probability plot indicates evidence of departures from normality in the residuals. Therefore we conclude that the regression assumptions are not verified. (b) (7.821, 8.437). We are 95% confident that the interval (13.5483, 14.6201) captures the population slope of the relationship between fat per gram and calories per gram. (c) Yes (d) . There is no relationship between fat per gram () and calories per gram . There is a linear relationship between fat per gram () and calories per gram (). Reject if . -value ≈ 0. Since the , we reject . There is evidence for a linear relationship between fat per gram () and calories per gram ().

Question 13.74

pulseandtemp

74. Open the Pulse and Temp data set. Estimate body temperature using heart rate .

Use technology for Exercises 75–78. Open the Crash data set, which contains information about the severity of injuries sustained by crash dummies when the National Transportation Safety Board crashed automobiles into a wall at 35 miles per hour.

Question 13.75

crash

75. The variable head_inj contains a measure of the severity of the head injury sustained by the dummies. The variable chest_in is a measure of the severity of the chest injury suffered by the crash dummies.

Would you expect a linear relationship to exist between the severity of head injuries and chest injuries?
Construct a scatterplot of the head_inj against chest_in. Describe the relationship between the variables.
If we were to perform a regression analysis using these two variables, is it clear which of the two variables we should label as the predictor and which we should label as the response? Explain.

13.1.75

(a) No (b) Positive relationship (c) Unclear

Question 13.76

crash

76. Perform a regression of the head injury severity on the chest injury severity .

What is the regression equation? Write it out in words and numbers, so that a nonstatistician would understand it.
Perform the appropriate hypothesis test, using level of significance .
Clearly interpret the meaning of the slope estimate .
Construct and interpret a 99% confidence interval for the true slope of the relationship between severity of head injury and severity of chest injury. How does your confidence interval support your conclusion in (b)?

Question 13.77

crash

77. The variable lleg_inj contains a measure of the severity of the injury sustained by the dummies' left legs. The variable weight contains the weight of the vehicles.

Would you expect a linear relationship to exist between the severity of left leg injuries and the weight of the vehicles?
Construct a scatterplot of the lleg_inj against weight. Describe the relationship between the variables.
If we were to perform a regression analysis using these two variables, is it clear which of the two variables we should label as the predictor and which we should label as the response? Explain.

13.1.77

(a) No (b) No apparent relationship between the variables (c) The weight of vehicles is the predictor variable and the severity of the leg injuries should be the response variable.

Question 13.78

crash

78. Perform a regression of the left leg injury severity on the vehicle weight .

What is the regression equation? Write it out in words and numbers, so that a nonstatistician would understand it.
Is the relationship significant? Perform the appropriate hypothesis test, using level of significance .
Construct and interpret a 95% confidence interval for the true slope of the relationship between vehicle weight and severity of left leg injury. How does your confidence interval support your conclusion in (b)?