Content check — Introductory Statistics, Chapter 13: F Distribution and One-Way ANOVA

13-1 §13.1 source ↗

Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. What are they?

Write one assumption.

Answer

Each population from which a sample is taken is assumed to be normal.

13-3 §13.1 source ↗

Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. What are they?

Write a third assumption.

Answer

The populations are assumed to have equal standard deviations (or variances).

13-5 §13.1 source ↗

Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. What are they?

Write the final assumption.

Answer

The response is a numerical value.

13-7 §13.1 source ↗

Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. What are they?

State the alternative hypothesis for a one-way ANOVA test if there are three groups.

Answer

H_a: At least two of the group means μ₁, μ₂, μ₃ are not equal.

13-9 §13.2 source ↗

Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in Table 13.13 are the weights for the different groups.

What is the Sum of Squares Factor?

Table 13.13 in the book ↗

Group 1	Group 2	Group 3
216	202	170
198	213	165
240	284	182
187	228	197
176	210	201

Answer

4,939.2

13-11 §13.2 source ↗

Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in Table 13.13 are the weights for the different groups.

What is the df for the numerator?

Table 13.13 in the book ↗

Group 1	Group 2	Group 3
216	202	170
198	213	165
240	284	182
187	228	197
176	210	201

Answer

2

13-13 §13.2 source ↗

Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in Table 13.13 are the weights for the different groups.

What is the Mean Square Factor?

Table 13.13 in the book ↗

Group 1	Group 2	Group 3
216	202	170
198	213	165
240	284	182
187	228	197
176	210	201

Answer

2,469.6

13-15 §13.2 source ↗

Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in Table 13.13 are the weights for the different groups.

What is the F statistic?

Table 13.13 in the book ↗

Group 1	Group 2	Group 3
216	202	170
198	213	165
240	284	182
187	228	197
176	210	201

Answer

3.7416

13-17 §13.2 source ↗

Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in Table 13.14.

What is the df for the numerator?

Table 13.14 in the book ↗

Team 1	Team 2	Team 3	Team 4
1	2	0	3
2	3	1	4
0	2	1	4
3	4	0	3
2	4	0	2

Answer

3

13-19 §13.2 source ↗

Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in Table 13.14.

What is SS_within?

Table 13.14 in the book ↗

Team 1	Team 2	Team 3	Team 4
1	2	0	3
2	3	1	4
0	2	1	4
3	4	0	3
2	4	0	2

Answer

13.2

13-21 §13.2 source ↗

Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in Table 13.14.

What is MS_within?

Table 13.14 in the book ↗

Team 1	Team 2	Team 3	Team 4
1	2	0	3
2	3	1	4
0	2	1	4
3	4	0	3
2	4	0	2

Answer

0.825

13-23 §13.2 source ↗

Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in Table 13.14.

Judging by the F statistic, do you think it is likely or unlikely that you will reject the null hypothesis?

Table 13.14 in the book ↗

Team 1	Team 2	Team 3	Team 4
1	2	0	3
2	3	1	4
0	2	1	4
3	4	0	3
2	4	0	2

Answer

Because a one-way ANOVA test is always right-tailed, a high F statistic corresponds to a low p-value, so it is likely that we will reject the null hypothesis.

13-25 §13.3 source ↗

What happens to the curves as the degrees of freedom for the numerator and the denominator get larger?

Answer

The curves approximate the normal distribution.

13-27 §13.3 source ↗

Use the following information to answer the next seven exercise. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in Table 13.15.

What is the df(denom)?

Table 13.15 in the book ↗

Team 1	Team 2	Team 3	Team 4	Team 5
36	32	48	38	41
42	35	50	44	39
51	38	39	46	40

Answer

ten

13-29 §13.3 source ↗

Use the following information to answer the next seven exercise. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in Table 13.15.

What are the Sum of Squares and Mean Squares Errors?

Table 13.15 in the book ↗

Team 1	Team 2	Team 3	Team 4	Team 5
36	32	48	38	41
42	35	50	44	39
51	38	39	46	40

Answer

SS = 237.33; MS = 23.73

13-31 §13.3 source ↗

Use the following information to answer the next seven exercise. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in Table 13.15.

What is the p-value?

Table 13.15 in the book ↗

Team 1	Team 2	Team 3	Team 4	Team 5
36	32	48	38	41
42	35	50	44	39
51	38	39	46	40

Answer

0.1614

13-33 §13.3 source ↗

Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Table 13.16

What is the df(num)?

Table 13.16 in the book ↗

Group A	Group B	Group C
101	151	101
108	149	109
98	160	198
107	112	186
111	126	160

Answer

two

13-35 §13.3 source ↗

Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Table 13.16

What are the SS_between and MS_between?

Table 13.16 in the book ↗

Group A	Group B	Group C
101	151	101
108	149	109
98	160	198
107	112	186
111	126	160

Answer

SS = 5,700.4;

MS = 2,850.2

13-37 §13.3 source ↗

Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Table 13.16

What is the F Statistic?

Table 13.16 in the book ↗

Group A	Group B	Group C
101	151	101
108	149	109
98	160	198
107	112	186
111	126	160

Answer

3.6101

13-39 §13.3 source ↗

Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Table 13.16

At the 10% significance level, are the scores among the different groups different?

Table 13.16 in the book ↗

Group A	Group B	Group C
101	151	101
108	149	109
98	160	198
107	112	186
111	126	160

Answer

Yes, there is enough evidence to show that the scores among the groups are statistically significant at the 10% level.

13-43 §13.4 source ↗

Use the following information to answer the next two exercises. There are two assumptions that must be true in order to perform an F test of two variances.

Name one assumption that must be true.

Answer

The populations from which the two samples are drawn are normally distributed.

13-45 §13.4 source ↗

Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.

State the null and alternative hypotheses.

Answer

H₀: σ₁ = σ₂

H_a: σ₁ < σ₂

or

H₀: $σ_{1}^{2} = σ_{2}^{2}$

H_a: $σ_{1}^{2} < σ_{2}^{2}$

13-47 §13.4 source ↗

Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.

What is s₂ in this problem?

Answer

4.11

13-49 §13.4 source ↗

Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.

What is the F statistic?

Answer

0.7159

13-51 §13.4 source ↗

Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.

Is the claim accurate?

Answer

No, at the 10% level of significance, we do not reject the null hypothesis and state that the data do not show that the variation in drive times for the first worker is less than the variation in drive times for the second worker.

13-53 §13.4 source ↗

Use the following information to answer the next four exercises. Two students are interested in whether or not there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are more consistent.

What is the F Statistic?

Answer

2.8674

13-55 §13.4 source ↗

Use the following information to answer the next four exercises. Two students are interested in whether or not there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are more consistent.

At the 5% significance level, do we reject the null hypothesis?

Answer

Reject the null hypothesis. There is enough evidence to say that the variance of the grades for the first student is higher than the variance in the grades for the second student.

13-57 §13.4 source ↗

Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records their speeds going up 35 hills. The first cyclist has a variance of 23.8 and the second cyclist has a variance of 32.1. The cyclists want to see if their variances are the same or different. Assume that commute times are normally distributed.

What is the F Statistic?

Answer

0.7414

13-59 §13.1 source ↗

Three different traffic routes are tested for mean driving time. The entries in the Table 13.18 are the driving times in minutes on the three different routes.

State SS_between, SS_within, and the F statistic.

Table 13.18 in the book ↗

Route 1	Route 2	Route 3
30	27	16
32	29	41
27	28	22
35	36	31

Answer

SS_between = 26
SS_within = 441
F = 0.2653

13-62 §13.2 source ↗

Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.

H₀: µ₁ = µ₂ = µ₃ = µ₄ = µ₅

Hα: At least any two of the group means µ₁, µ₂, …, µ₅ are not equal.

degrees of freedom – denominator: df(denom) = ________

Table 13.20 in the book ↗

	Northeast	South	West	Central	East
	16.3	16.9	16.4	16.2	17.1
	16.1	16.5	16.5	16.6	17.2
	16.4	16.4	16.6	16.5	16.6
	16.5	16.2	16.1	16.4	16.8
$\bar{x} =$
$s^{2} =$

Answer

df(denom) = 15

13-64 §13.3 source ↗

DIRECTIONS

Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E Solution Sheets.

Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat's weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again, and the net gain in grams is recorded. Using a significance level of 10%, test the hypothesis that the three formulas produce the same mean weight gain.

Table 13.21 Weights of Student Lab Rats in the book ↗

Linda's rats	Tuan's rats	Javier's rats
43.5	47.0	51.2
39.4	40.5	40.9
41.3	38.9	37.9
46.0	46.3	45.0
38.2	44.2	48.6

Answer

a. H₀: µ_L = µ_T = µ_J

b. H_a: at least any two of the means are different

c. df(num) = 2; df(denom) = 12

d. F distribution

e. 0.67

f. 0.5305

g. Answers may vary.

h. Decision: Do not reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the means are different.

13-67 §13.3 source ↗

DIRECTIONS

Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E Solution Sheets.

Eliminate one magazine type that you now feel has a mean length different from the others. Redo the hypothesis test, testing that the remaining three means are statistically the same. Use a new solution sheet. Based on this test, are the mean lengths for the remaining three magazines statistically the same?

Answer

a. H_a: µ_c = µ_n = µ_h

b. At least any two of the magazines have different mean lengths.

c. df(num) = 2, df(denom) = 12

d. F distribtuion

e. F = 15.28

f. p-value = 0.0005

g. Answers may vary.

h.

i. Alpha: 0.05

ii. Decision: Reject the Null Hypothesis.

iii. Reason for decision: p-value < alpha

iv. Conclusion: There is sufficient evidence to conclude that the mean lengths of the magazines are different.

13-69 §13.3 source ↗

DIRECTIONS

Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E Solution Sheets.

Are the means for the final exams the same for all statistics class delivery types? Table 13.25 shows the scores on final exams from several randomly selected classes that used the different delivery types.

Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.

Table 13.25 in the book ↗

Online	Hybrid	Face-to-Face
72	83	80
84	73	78
77	84	84
80	81	81
81		86
		79
		82

Answer

a. H₀: μ_o = μ_h = μ_f

b. At least two of the means are different.

c. df(n) = 2, df(d) = 13

d. F_2,13

e. 0.64

f. 0.5437

g. Answers may vary.

h.

i. Alpha: 0.05

ii. Decision: Do not reject the null hypothesis.

iii. Reason for decision: p-value > alpha

iv. Conclusion: The mean scores of different class delivery are not different.

13-71 §13.3 source ↗

DIRECTIONS

Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E Solution Sheets.

Are the mean numbers of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that Table 13.27 shows the results of a study.

Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.

Table 13.27 in the book ↗

Powder	Machine Made	Hard Packed
1,210	2,107	2,846
1,080	1,149	1,638
1,537	862	2,019
941	1,870	1,178
	1,528	2,233
	1,382

Answer

a. H₀: μ_p = μ_m = μ_h

b. At least any two of the means are different.

c. df(n) = 2, df(d) = 12

d. F_2,12

e. 3.13

f. 0.0807

g. Answers may vary.

h.

i. Alpha: 0.05

ii. Decision: Do not reject the null hypothesis.

iii. Reason for decision: p-value > alpha

iv. Conclusion: There is not sufficient evidence to conclude that the mean numbers of daily visitors are different.

13-73 §13.3 source ↗

Table 13.29 in the book ↗

RS	SS	NS	RS	SS	NS
12.8	38.4	35.4	22.4	23.1	22.6
21.6	32.9	27.4	27.5	29.4	40.4
14.8	48.5	19.3	20.3	16	34.4
23.1	20.9	41.8	38.7	20.1	30.4
34.6	11.6	20.3	26.4	23.3	14.9
19.7	22.3	37.6	23.7	22.9	51.8
22.6	30.2	36.9	26.1	22.5	33.8
29.6	33.4	37.3	29.5	15.1	37.9
16.4	26.7	28.2	38.6	31	29.5
20.3	39	23.4	44.4	16.9	42.4
29.3	12.8	33.7	23.2	16.1	36.6
14.9	14.6	29.2	23.6	10.8	47.4
27.3	12.2	41.7

This graph is a scatterplot which represents the data provided. The horizontal axis is labeled 'Mean eggs laid per day' and extends from 10 - 50. The vertical axis is labeled 'Fruitflies DDT resistant or susceptible, or not selected.' The vertical axis is labeled with the categories NS, RS, SS. — Figure 13.8 in the book ↗

DIRECTIONS

Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E Solution Sheets.

DDT is a pesticide that has been banned from use in the United States and most other areas of the world. It is quite effective, but persisted in the environment and over time became seen as harmful to higher-level organisms. Famously, egg shells of eagles and other raptors were believed to be thinner and prone to breakage in the nest because of ingestion of DDT in the food chain of the birds.

An experiment was conducted on the number of eggs (fecundity) laid by female fruit flies. There are three groups of flies. One group was bred to be resistant to DDT (the RS group). Another was bred to be especially susceptible to DDT (SS). Finally there was a control line of non-selected or typical fruitflies (NS). Here are the data:

The values are the average number of eggs laid daily for each of 75 flies (25 in each group) over the first 14 days of their lives. Using a 1% level of significance, are the mean rates of egg selection for the three strains of fruitfly different? If so, in what way? Specifically, the researchers were interested in whether or not the selectively bred strains were different from the nonselected line, and whether the two selected lines were different from each other.

Here is a chart of the three groups:

Answer

The data appear normally distributed from the chart and of similar spread. There do not appear to be any serious outliers, so we may proceed with our ANOVA calculations, to see if we have good evidence of a difference between the three groups.

H₀: μ₁ = μ₂ = μ₃;

H_a: μ_i ≠ μ_j some i ≠ j.

Define μ₁, μ₂, μ₃, as the population mean number of eggs laid by the three groups of fruit flies.

F statistic = 8.6657;

p-value = 0.0004

Decision: Since the p-value is less than the level of significance of 0.01, we reject the null hypothesis.

Conclusion: We have good evidence that the average number of eggs laid during the first 14 days of life for these three strains of fruitflies are different.

Interestingly, if you perform a two sample t-test to compare the RS and NS groups they are significantly different (p = 0.0013). Similarly, SS and NS are significantly different (p = 0.0006). However, the two selected groups, RS and SS are not significantly different (p = 0.5176). Thus we appear to have good evidence that selection either for resistance or for susceptibility involves a reduced rate of egg production (for these specific strains) as compared to flies that were not selected for resistance or susceptibility to DDT. Here, genetic selection has apparently involved a loss of fecundity.

This graph shows a nonsymmetrical F distribution curve. This curve does not have a peak, but slopes downward from a maximum value at (0, 1.0) and approaches the horiztonal axis at the right edge of the graph. — Figure 13.9 in the book ↗

13-75 §13.4 source ↗

Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded.

Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats. Test at a significance level of 10%.

Table 13.31 in the book ↗

Linda's rats	Tuan's rats	Javier's rats
43.5	47.0	51.2
39.4	40.5	40.9
41.3	38.9	37.9
46.0	46.3	45.0
38.2	44.2	48.6

Answer

a. $H_{0} : σ_{1}^{2} =$ $σ_{2}^{2}$

b. $H_{a} : σ_{1}^{2} \neq σ_{1}^{2}$

c. df(num) = 4; df(denom) = 4

d. F_{4, 4}

e. 3.00

f. 2(0.1563) = 0.3126. Using the TI-83+/84+ function 2-SampFtest, you get the test statistic as 2.9986 and p-value directly as 0.3127. If you input the lists in a different order, you get a test statistic of 0.3335 but the p-value is the same because this is a two-tailed test.

g. Answers may vary.

h. Decision: Do not reject the null hypothesis; Conclusion: There is insufficient evidence to conclude that the variances are different.

13-78 §13.4 source ↗

Which two magazine types do you think have different variances in length?

Answer

The answers may vary. Sample answer: Home decorating magazines and news magazines have different variances.

13-80 §13.4 source ↗

Are the variances for incomes on the East Coast and the West Coast the same? Suppose that Table 13.35 shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of 0.05.

Table 13.35 in the book ↗

East	West
38	71
47	126
30	42
82	51
75	44
52	90
115	88
67

Answer

a. H₀: = $σ_{1}^{2}$ = $σ_{2}^{2}$

b. H_a: $σ_{1}^{2}$ ≠ $σ_{1}^{2}$

c. df(n) = 7, df(d) = 6

d. F_7,6

e. 0.8117

f. 0.7825

g. Answers may vary.

h.

i. Alpha: 0.05

ii. Decision: Do not reject the null hypothesis.

iii. Reason for decision: p-value > alpha

iv. Conclusion: There is not sufficient evidence to conclude that the variances are different.

13-82 §13.4 source ↗

King Manuel I, Komnenus ruled the Byzantine Empire from Constantinople (Istanbul) during the years 1145 to 1180 A.D. The empire was very powerful during his reign, but declined significantly afterwards. Coins minted during his era were found in Cyprus, an island in the eastern Mediterranean Sea. Nine coins were from his first coinage, seven from the second, four from the third, and seven from a fourth. These spanned most of his reign. We have data on the silver content of the coins:

Did the silver content of the coins change over the course of Manuel’s reign?

Here are the means and variances of each coinage. The data are unbalanced.

Table 13.37 in the book ↗

First Coinage	Second Coinage	Third Coinage	Fourth Coinage
5.9	6.9	4.9	5.3
6.8	9.0	5.5	5.6
6.4	6.6	4.6	5.5
7.0	8.1	4.5	5.1
6.6	9.3		6.2
7.7	9.2		5.8
7.2	8.6		5.8
6.9
6.2

Table 13.38 in the book ↗

	First	Second	Third	Fourth
Mean	6.7444	8.2429	4.875	5.6143
Variance	0.2953	1.2095	0.2025	0.1314

Answer

Here is a strip chart of the silver content of the coins:

While there are differences in spread, it is not unreasonable to use ANOVA techniques. Here is the completed ANOVA table:

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F
Factor (Between)	37.748	4 – 1 = 3	12.5825	26.272
Error (Within)	11.015	27 – 4 = 23	0.4789
Total	48.763	27 – 1 = 26

Table 13.42

P(F > 26.272) = 0;

Reject the null hypothesis for any alpha. There is sufficient evidence to conclude that the mean silver content among the four coinages are different. From the strip chart, it appears that the first and second coinages had higher silver contents than the third and fourth.

This graph is a scatterplot which represents the data provided. The horizontal axis is labeled 'Silver content coins' and extends from 5 - 9. The vertical axis is labeled 'Coinage.' The vertical axis is labeled with the categories First, Second, Third, and Fourth. — Figure 13.10 in the book ↗

Table 13.42 in the book ↗

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F
Factor (Between)	37.748	4 – 1 = 3	12.5825	26.272
Error (Within)	11.015	27 – 4 = 23	0.4789
Total	48.763	27 – 1 = 26

13-83 §13.4 source ↗

The American League and the National League of Major League Baseball are each divided into three divisions: East, Central, and West. Many years, fans talk about some divisions being stronger (having better teams) than other divisions. This may have consequences for the postseason. For instance, in a certain year Tampa Bay won 90 games and did not play in the postseason, while Detroit won only 88 and did play in the postseason. This may have been an oddity, but is there good evidence that in this year, the American League divisions were significantly different in overall records? Use the following data to test whether the mean number of wins per team in the three American League divisions were the same or not. Note that the data are not balanced, as two divisions had five teams, while one had only four.

Table 13.39 in the book ↗

Division	Team	Wins
East	NY Yankees	95
East	Baltimore	93
East	Tampa Bay	90
East	Toronto	73
East	Boston	69

Table 13.40 in the book ↗

Division	Team	Wins
Central	Detroit	88
Central	Chicago Sox	85
Central	Kansas City	72
Central	Cleveland	68
Central	Minnesota	66

Table 13.41 in the book ↗

Division	Team	Wins
West	Oakland	94
West	Texas	93
West	LA Angels	89
West	Seattle	75

Answer

Here is a stripchart of the number of wins for the 14 teams in the AL for the 2012 season.

While the spread seems similar, there may be some question about the normality of the data, given the wide gaps in the middle near the 0.500 mark of 82 games (teams play 162 games each season in MLB). However, one-way ANOVA is robust.

Here is the ANOVA table for the data:

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F
Factor (Between)	344.16	3 – 1 = 2	172.08
Error (Within)	1,219.55	14 – 3 = 11	110.87	1.5521
Total	1,563.71	14 – 1 = 13

Table 13.43

P(F > 1.5521) = 0.2548
Since the p-value is so large, there is not good evidence against the null hypothesis of equal means. We decline to reject the null hypothesis. Thus, for 2012, there is not any have any good evidence of a significant difference in mean number of wins between the divisions of the American League.

This graph is a scatterplot which represents the data provided. The horizontal axis is labeled 'Number of wins in 2012 Major League Baseball Season' and extends from 65 - 95. The vertical axis is labeled 'American league division.' The vertical axis is labeled with the categories Central, East, West. — Figure 13.11 in the book ↗

Table 13.43 in the book ↗

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F
Factor (Between)	344.16	3 – 1 = 2	172.08
Error (Within)	1,219.55	14 – 3 = 11	110.87	1.5521
Total	1,563.71	14 – 1 = 13

First Coinage	Second Coinage	Third Coinage	Fourth Coinage
5.9	6.9	4.9	5.3
6.8	9.0	5.5	5.6
6.4	6.6	4.6	5.5
7.0	8.1	4.5	5.1
6.6	9.3		6.2
7.7	9.2		5.8
7.2	8.6		5.8
6.9
6.2

First Coinage	Second Coinage	Third Coinage	Fourth Coinage
5.9	6.9	4.9	5.3
6.8	9.0	5.5	5.6
6.4	6.6	4.6	5.5
7.0	8.1	4.5	5.1
6.6	9.3		6.2
7.7	9.2		5.8
7.2	8.6		5.8
6.9
6.2

First Coinage	Second Coinage	Third Coinage	Fourth Coinage
5.9	6.9	4.9	5.3
6.8	9.0	5.5	5.6
6.4	6.6	4.6	5.5
7.0	8.1	4.5	5.1
6.6	9.3		6.2
7.7	9.2		5.8
7.2	8.6		5.8
6.9
6.2