Section 12.1 Exercises

CLARIFYING THE CONCEPTS

Question 12.1

1. Does the overall sample mean always equal the mean of the sample means? Explain. (p. 670)

12.1.1

No. If the sample sizes are not all the same, then we need to calculate the overall sample mean by calculating the weighted mean of the sample means where the weights are the sample sizes.

Question 12.2

2. What does MSTR measure? What does MSE measure? (p. 671)

Question 12.3

3. In your own words, explain how ANOVA works. (p. 668)

12.1.3

Answers will vary.

Question 12.4

4. What are the required conditions for performing an analysis of variance? (p. 668)

Question 12.5

5. A comparison dotplot of the SAT scores of three sororities shows no overlap at all between the groups. Does this represent evidence for or against the null hypothesis that all population means are equal? (p. 667)

12.1.5

Against.

Question 12.6

6. True or false: If we reject the null hypothesis in an analysis of variance, then there is evidence that all the population mean sizes are different. If the statement is false, explain why it is false. (p. 673)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 7–10 Example 1 Verifying the requirements
for ANOVA
Exercises 11–14 Example 2 Calculating the overall
sample mean
Exercises 15–22 Example 3 Constructing the
ANOVA table
Exercises 23–28 Example 4 One-way ANOVA:
-value method
Exercises 29–30 Example 6 One-way ANOVA:
critical-value method

For Exercises 7–10, assume that the samples were drawn independently and confirm that the other requirements for ANOVA are met by doing the following:

  1. Use technology to construct a probability plot for each group. Check for normality.
  2. Check for equal variances.

Question 12.7

7.

A B C
5 10 15
10 15 20
15 20 25

12.1.7

(a)

image

image

image

All three graphs show acceptable normality.

(b)

image

All three standard deviations are equal to 5, so all of them are less than . Therefore, the equal variance requirement is satisfied.

Question 12.8

8.

A B C
10 12 14
20 18 20
30 30 26

Question 12.9

9.

A B C D
30 35 45 5
0 75 20 30
25 55 10 0
0 45 25 35

12.1.9

(a)

image

image

image

image

All four graphs show acceptable normality.

(b)

image

The smallest standard deviation is 14.72 which is the standard deviation for group C. Twice 14.72 is 29.44. All other standard deviations are less than 29.44 so the equal variance requirement is satisfied.

Question 12.10

10.

A B C D
5 8 3 6
2 7 7 4
4 10 5 5
1 6 4 3
3 9 6 7

682

For Exercises 11–14, calculate the overall sample mean for the indicated data.

Question 12.11

11. Data in Exercise 7

12.1.11

Question 12.12

12. Data in Exercise 8

Question 12.13

13. Data in Exercise 9

12.1.13

Question 12.14

14. Data in Exercise 10

For Exercises 15–22, do the following for the indicated data.

  1. Calculate the following:
  2. Construct the ANOVA table using the statistics computed in (a).

Question 12.15

15. Data in Exercise 7

12.1.15

(a) (i) SSTR = 150 (ii) SSE = 150 (iii) SST = 300 (iv) (v) (vi) MSTR = 75 (vii) MSE = 25

(b)

image

Question 12.16

16. Data in Exercise 8

Question 12.17

17. Data in Exercise 9

12.1.17

(a) (i) SSTR = 3680 (ii) SSE = 3219 (iii) SST = 6898 (iv) (v) (vi) MSTR = 1226.6 (vii) MSE = 268.2

(b)

image

Question 12.18

18. Data in Exercise 10

Question 12.19

19.

Sample A Sample B Sample C

12.1.19

(a) (i) SSTR = 40 (ii) SSE = 12 (iii) SST = 52 (iv) (v) (vi) MSTR = 20 (vii) MSE = 1

(b)

Source of
variation		 Sum of
squares		 Degrees of
freedom		 Mean
square
Treatments 40 2 20 20
Error 12 12 1
Total 52

Question 12.20

20.

Sample A Sample B Sample C Sample D

Question 12.21

21.

Sample A Sample B Sample C Sample D

12.1.21

(a) (i) SSTR = 395,833.3333 (ii) SSE = 15,748 (iii) SST = 411,581.3333 (iv) (v) (vi) MSTR = 131,944.4444 (vii) MSE = 26.4228

(b)

Source of
variation		 Sum of
squares		 Degrees of
freedom		 Mean
square
Treatments 395,833.3333 3 131,944.4444 4993.5794
Error 15,748 594 26.4228
Total 411,581.3333

Question 12.22

22.

Sample A Sample B Sample C Sample D

For Exercises 23–30, assume that the ANOVA assumptions are verified.

For the ANOVA table indicated in Exercises 23–28, test whether the population means differ, using the -value method and .

Question 12.23

23. ANOVA table from Exercise 15

12.1.23

versus Not all the population means are equal.

Reject if the .

.

.

The is not . Therefore we do not reject . There is not enough evidence to conclude at level of significance that the population means are not all equal.

Question 12.24

24. ANOVA table from Exercise 16

Question 12.25

25. ANOVA table from Exercise 17

12.1.25

versus Not all the population means are equal.

Reject if the .

.

.

The is . Therefore we reject . There is evidence at level of significance that the population means are not all equal.

Question 12.26

26. ANOVA table from Exercise 18

Question 12.27

27. ANOVA table from Exercise 19

12.1.27

: Not all the population means are equal.

Reject if the .

.

.

The is . Therefore we reject . There is evidence at level of significance that the population means are not all equal.

Question 12.28

28. ANOVA table from Exercise 20

For the data indicated in Exercises 29–30, test whether the population means differ, using the critical-value method and .

Question 12.29

29. ANOVA table from Exercise 21

12.1.29

: Not all the population means are equal.

Reject if ;

;

is . Therefore we reject . There is evidence at level of significance that the population means are not all equal.

Question 12.30

30. ANOVA table from Exercise 22

APPLYING THE CONCEPTS

For Exercises 31–34, assume that the data are independently drawn random samples from normal populations.

  1. Verify the equal-variance assumption.
  2. Calculate the following measures:
    1. and
    3. SSTR
    4. SSE
    5. SST
    6. MSTR
    7. MSE
  3. Construct the ANOVA table.
  4. Perform the appropriate one-way ANOVA using level of significance .

Question 12.31

cafeanova

31. Student-Run Café Business. In Example 8 in Chapter 2, we looked at data from a student-run café business. The table contains the number of food items sold per day.

Wraps Muffins Chips
12 6 7
13 3 16
19 10 8
5 1 4
22 8 10

12.1.31

(a)

image

The smallest standard deviation is 3.65, which is the standard deviation for muffins. Twice 3.65 is 7.30. All other standard deviations are less than 7.30, so the equal variance requirement is satisfied.

(b) (i) , (ii) (iii) SSTR = 187.6 (iv) SSE = 308 (v) SST = 495.6 (vi) MSTR = 93.80 (vii) MSE = 25.67 (viii)

(c)

image

(d) ;

: Not all the population means are equal.

Reject if the .

.

.

The is not . Therefore we do not reject . There is not enough evidence to conclude at level of significance that the population means of the number of different types of food items are not all equal.

Question 12.32

prosdartsanova

32. The professionals versus the Darts. In the Chapter 3 Case Study, we examined stock market returns for professional financial analysts, compared to random darts and the Dow Jones Industrial Average (DJIA). The table contains daily stock market returns.

Pros Darts DJIA
10.6 20.6 4.4
27.8 18.5 11.2
29.1 1.8 3.7
2.2 11.7 17.6
14.1 1.8 0.2

Question 12.33

weightages

33. Weight and Age. The Chapter 4 Case Study looked at body measurements for physically fit males and females. Is there a difference in weight among different age groups? The table contains the weights of five randomly chosen females from each of three age groups: younger (18-22), middle (23-30), and older (31+).

Younger Middle Older
119 142.9 121.3
124.8 104.3 107.4
130.7 115.1 169.3
130.1 98.8 122.8
155.4 110.2 155.4

12.1.33

(a)

image

The smallest standard deviation is 13.91, which is the standard deviation for the younger. Twice 13.91 is 27.82. All other standard deviations are less than 27.82, so the equal variance requirement is satisfied.

(b) (i) , (ii) (iii) SSTR = 1276 (iv) SSE = 4640 (v) SST = 5916 (vi) MSTR = 637.8 (vii) MSE = 386.7 (viii)

(c)

image

(d) ;

: Not all the population means are equal.

Reject if the .

.

.

The is not . Therefore we do not reject . There is not enough evidence to conclude at level of significance that the population mean weights of the different age groups are not all equal.

683

Question 12.34

fullmoon

34. The Full Moon and Emergency Room Visits. Is there a difference in emergency room visits before, during, and after a full moon? A study looked at the admission rate (number of patients per day) to the emergency room of a Virginia mental health clinic over a series of 12 full moons.2 The data are provided in the table. Is there evidence of a difference in emergency room visits before, during, and after the full moon?

Before During After
6.4 11.5 5 13 5.8 13.5
7.1 13.8 13 16 9.2 13.1
6.5 15.4 14 25 7.9 15.8
8.6 15.7 12 14 7.7 13.3
8.1 11.7 6 14 11.0 12.8
10.4 15.8 9 20 12.9 14.5

Question 12.35

35. Missing Values. Part of an ANOVA table for an analysis of variance involving 7 groups for a study follows; each sample contained 10 data values.

Source of
variation		 Sum of
square		 Degrees of
freedom		 Mean
squares
Treatment 120 — — —
Error 315 — —
Total —
  1. Find all six missing values in the table and fill in the blanks.
  2. Perform the appropriate hypothesis test using .

12.1.35

(a) Missing values are in red.

Source of
variation		 Sum of
squares		 Degrees of
freedom		 Mean
square
Treatments 120 6 20 4
Error 315 63 5
Total 435

(b) . : Not all the population means are equal. Reject if the . . Since the , we reject . There is evidence that not all the population means are equal.

Question 12.36

36. Missing Values. Part of an ANOVA table for an analysis of variance involving three groups follows; each sample contained six data values.

Source of
variation		 Sum of
square		 Degrees of
freedom		 Mean
squares
Treatment — — — —
Error 90 — —
Total 150
  1. Find all six missing values in the table.
  2. Perform the appropriate hypothesis test using .

Question 12.37

37. Missing Values. Part of an ANOVA table follows.

Source of
variation		 Sum of
square		 Degrees of
freedom		 Mean
squares
Treatment — 4 10 1.0
Error — — —
Total 440
  1. Find all four missing values in the table and fill in the blanks.
  2. Perform the appropriate hypothesis test using .

12.1.37

(a) Missing values are in red.

Source of
variation		 Sum of
squares		 Degrees of
freedom		 Mean
square		 -test
statistic
Treatments SSTR = 40 MSTR = 10
Error SSE = 400 MSE = 10
Total SST = 440

(b) . : Not all the population means are equal. . Reject if . . Since is not we do not reject . There is insufficient evidence that not all the population means are equal.

Question 12.38

38. Missing Values. Part of an ANOVA table follows.

Source of
variation		 Sum of
square		 Degrees of
freedom		 Mean
squares
Treatment — 2 — 2.0
Error 480 — 24
Total —
  1. Find all four missing values in the table and fill in the blanks.
  2. Perform the appropriate hypothesis test using .

Question 12.39

39. ANOVA Can Be Applied to Two populations. Researchers are interested in whether the mean heart rates of women and men differ. The following table provides summary statistics of random samples of pulse rates drawn from groups of women and men.

  1. Test, using , whether the population mean pulse rates differ.
  2. Which method of inference from an earlier chapter could we also use to solve this problem?
Females Males
65 65
98.384 98.104
0.743 0.699

12.1.39

(a) . : The two population means are not equal. = the population mean heart rate for females; = the population mean heart rate for males. Reject if the . , . Since the is , we reject . There is evidence that the population mean heart rates are not equal.

(b) Inference for Two Independent Means, Section 10.2.

Do Taller people Have lower Voices? Use this information for Exercises 40 and 41. The heights of the members of the New York Choral Society were recorded.3 Choral singers are roughly divided into four groups, from highest-pitched voices to lowest: sopranos and altos (usually female), and tenors and basses (usually male). Assume that the assumptions are satisfied.

Question 12.40

40. What Result Might We Expect? The comparison boxplot of the heights is shown here. Is there evidence for differences in the population mean heights?

image

Question 12.41

41. The summary statistics are shown here. Test, at , whether the population mean heights of the groups differ.

image

12.1.41

Since the is , we reject . There is evidence that not all the population mean heights are equal.

684

Nutritional Rating of Breakfast Cereals. Use this information for Exercises 42–45. Does the nutritional rating of breakfast cereals vary by manufacturer? The data set Breakfast Cereals 2 contains information on the nutritional rating and manufacturer for 68 breakfast cereals.4 Assume that the cereals were selected randomly from the manufacturers so that the selection of a cereal from one manufacturer did not affect the selection of cereals from other manufacturers.

Question 12.42

breakfastcereals2

42. What Result Might We Expect? The comparison boxplot of the nutritional ratings is shown here. Assume that the normality condition is satisfied.

image
  1. Explain whether the plot may be considered evidence for or against the null hypothesis that all population mean nutritional ratings were equal.
  2. Would we expect the null hypothesis to be rejected or not to be rejected?

Question 12.43

breakfastcereals2

43. The largest standard deviation is , whereas the smallest is . State whether the equal-variance requirement is satisfied.

12.1.43

Since , the equal variance requirement is satisfied.

Question 12.44

breakfastcereals2

44. Is the independence assumption validated? Why?

Question 12.45

breakfastcereals2

45. The Minitab output shown here contains the ANOVA results. and the -value are in blue. Test, using level of significance , whether all population mean nutritional ratings are equal across all manufacturers.

image

12.1.45

: Not all the population means are equal. Reject if the . ; . Since the is , we reject . There is evidence that the population mean nutritional ratings of the cereals made by the different manufacturers are not all equal.

Question 12.46

image 46. Student-Run Café Business. Try to answer these questions without redoing the calculations and without rerunning the ANOVA. Refer to Exercise 31. What if more than 22 wraps were sold instead of the last entry for 22 recorded for the number of wraps sold?

  1. How would that change affect each entry in the ANOVA table?
  2. How would that change affect the conclusion, if at all?

Question 12.47

image 47. Challenge Exercise: The professionals versus the Darts. Try to answer these questions without redoing the calculations and without rerunning the ANOVA. Refer to Exercise 32. What if we added three points to each value for the DJIA? (Hint: The sample mean for the DJIA is more than three points lower than the other two sample means.)

  1. How would that change affect each entry in the ANOVA table?
  2. How would that change affect the conclusion, if at all?

12.1.47

(a) (i) and would stay the same (ii) would increase (iii) SSTR would decrease (iv) SSE would stay the same (v) SST would decrease (vi) MSTR would decrease (vii) MSE would stay the same (viii) would decrease (ix) the would increase. (b) would decrease and the would increase. Therefore, the conclusion would still be do not reject .

BRINGING IT ALL TOGETHER

Gas Mileage for European, Japanese, and American Cars. Use this information for Exercises 48–50. The following figure shows a comparison boxplot of the vehicle mileage (in mpg) for random samples of automobiles manufactured in Europe, Japan, and the United States. The summary statistics are provided. We are interested in testing, using , whether population mean gas mileage differs among automobiles from the three regions. Assume that the assumptions are satisfied.

image
MPG Sample 1:
Europe		 Sample 2:
Japan		 Sample 3:
USA
Sample mean
Sample standard
deviation
Sample size

Question 12.48

48. What Result Might We Expect?

  1. Based on the graphical evidence in the comparison boxplot, what might be the conclusion? Explain your reasoning.
  2. Perform the ANOVA, using whichever method you prefer.
  3. Is your intuition from (a) supported?

Question 12.49

49. Confidence Intervals as Further Clues in ANOVA. Refer to Exercise 48. Suppose we construct a confidence interval for each of the population means. If at least one confidence interval does not overlap the others, then it is evidence against the null hypothesis.

685

  1. Use a interval from Section 8.2 to construct a 99% confidence interval for the population mean gas mileage of
    1. European cars
    2. Japanese cars
    3. American cars
  2. Is there one confidence interval from (a) that does not overlap the other two? If so, what does this mean in terms of the null hypothesis that all the population means are equal?

12.1.49

(a) (i) (25.4805, 29.7255) (ii) (28.6366, 32.2654) (iii) (18.9526, 21.1134). (b) The confidence interval for the population mean gas mileage of American cars does not overlap the other two confidence intervals. This is evidence against the null hypothesis that all the population means are equal.

Question 12.50

image 50. Refer to the table of descriptive summaries of vehicle mileage. What if we discovered that we made a mistake in the data collection and that every Japanese vehicle tested actually had 1 mpg higher gas mileage than previously recorded? Explain how and why this change would affect the following measures—increase, decrease, or no change.

  3. SSTR
  4. SSE
  5. SST
  6. MSTR
  7. MSE
  9. -value
  10. Conclusion

WORKING WITH LARGE DATA SETS

For Exercises 51–54, use 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 12.51

crash

51. Head Injuries and Vehicle Size. The variable head_inj contains a measure of the severity of the head injury sustained by crash dummies. The variable size2 categorizes the type of vehicle, such as light, medium, heavy, pickup truck, MPV (SUV), and so on. The values of the variable size2 are as follows: 1 = compact car, 2 = light car, 3 = medium car, 4 = heavy car, = = minicompact car, 6 = van, 7 = pickup truck, and 8 = MPV (SUV). Would you expect the population mean severity of head injuries suffered by the dummies to be the same across all the size categories? Use technology to perform the analysis of variance, using . Comment on the results.

12.1.51

.: Not all the population means are equal. Reject if the . . Since the is , we reject . There is evidence that not all the population means are equal.

image

Question 12.52

crash

52. Head Injuries and Vehicle year. The variable year indicates the model year of the vehicle. Would you expect the population mean severity of head injuries suffered by the dummies to be the same across all model years? Use technology to perform the analysis of variance, using . Comment on the results.

Question 12.53

crash

53. Chest Injury and protection Device. The variable chest_in is a measure of the severity of the chest injury suffered by the crash dummy. The variable protect2 categorizes the type of protection that the vehicle offers: 1 = manual seat belts, 2 = motorized seat belts, 3 = passive seat belts, 4 = dual air bags. Would you expect the population mean severity of chest injuries suffered by the dummies to be the same for all categories of protection? Use technology to perform the analysis of variance, using . Comment on the results.

12.1.53

.: Not all the population means are equal. Reject if the . . Since the is not , we do not reject . There is insufficient evidence that not all the population means are equal.

image

Question 12.54

crash

54. Refer to Exercise 53. Now use . How does this affect your conclusion?

Use the Analysis of Variance applet for Exercises 55 and 56.

Question 12.55

55. Move the group means so that they are about the same by clicking and dragging the black dots so that they are about even horizontally.

  1. What happens to the value of ?
  2. Explain why this happens, using the concept of between-sample variability and the statistics SSTR, MSTR, and

12.1.55

(a) Decreases (b) Since all of the sample means are changed until they are about the same, SSTR and MSTR should decrease to small numbers. Since the variability of each sample stays the same, MSE stays the same. Thus decreases.

Question 12.56

56. Click Reset. Increase the Within-Sample Variability.

  1. What happens to the value of ?
  2. Explain why this change happens, using the concept of within-sample variability and the statistics SSE, MSE, and .