Analysis of Variance 11-1
CHAPTER 11: ANALYSIS OF VARIANCE
1. In a one-way ANOVA, if the computed F statistic is greater than the critical F value you may
a) reject H0 since there is evidence all the means differ.
b) reject H0 since there is evidence that not all the means are different.
c) not reject H0 since there is no evidence of a difference in the means.
d) not reject H0 because a mistake has been made.
2. Which of the following components in an ANOVA table are not additive?
a) Sum of squares.
b) Degrees of freedom.
c) Mean squares.
d) It is not possible to tell.
3. When would you use the Tukey-Kramer procedure?
a) To test for normality.
b) To test for homogeneity of variance.
c) To test independence of errors.
d) To test for differences in pairs of means.
4. A completely randomized design
a) has only one factor with several treatment groups.
b) can have more than one factor, each with several treatment groups.
c) has one factor and one block.
d) has one factor and one block and multiple values.
11-2 Analysis of Variance
5. The F test statistic in a one-way ANOVA is
a) MSW/MSA.
b) SSW/SSA.
c) MSA/MSW.
d) SSA/SSW.
6. The degrees of freedom for the F test in a one-way ANOVA are
a) (n – c) and (c – 1).
b) (c – 1) and (n – c).
c) (c – n) and (n – 1).
d) (n – 1) and (c – n).
7. In a one-way ANOVA, the null hypothesis is always
a) there is no difference in the population means.
b) there is some treatment effect.
c) all the population means are different.
d) some of the population means are different.
8. In a one-way ANOVA
a) an interaction term is present.
b) an interaction effect can be tested.
c) there is no interaction term.
d) the interaction term has (c – 1)(n – 1) degrees of freedom.
Analysis of Variance 11-3
9. An airline wants to select a computer software package for its reservation system. Four software
packages (1, 2, 3, and 4) are commercially available. The airline will choose the package that
bumps the fewest mean number of passengers as possible during a month. An experiment is set
up in which each package is used to make reservations for 5 randomly selected weeks. (A total of
20 weeks was included in the experiment.) The number of passengers bumped each week is
given below. How should the data be analyzed?
Package 1: 12, 14, 9, 11, 16
Package 2: 2, 4, 7, 3, 1
Package 3: 10, 9, 6, 10, 12
Package 4: 7, 6, 6, 15, 12
a) F test for differences in variances.
b) One-way ANOVA F test.
c) t test for the differences in means.
d) t test for the mean difference.
10. True or False: The analysis of variance (ANOVA) tests hypotheses about the population
variance.
11. True or False: When the F test is used for ANOVA, the rejection region is always in the right
tail.
12. True or False: If you are comparing the mean sales among 3 different brands you are dealing
with a three-way ANOVA design.
11-4 Analysis of Variance
13. True or False: The MSE must always be positive.
14. True or False: In a one-factor ANOVA analysis, the among sum of squares and within sum of
squares must add up to the total sum of squares.
SCENARIO 11-1
An airline wants to select a computer software package for its reservation system. Four software
packages (1, 2, 3, and 4) are commercially available. The airline will choose the package that bumps
as few passengers as possible during a month. An experiment is set up in which each package is used
to make reservations for 5 randomly selected weeks. (A total of 20 weeks was included in the
experiment.) The number of passengers bumped each week is obtained, which gives rise to the
following Excel output:
ANOVA
Source of Variation
SS
df
MS
F
P-value
F crit
Between Groups
212.4
3
8.304985
0.001474
3.238867
Within Groups
136.4
8.525
Total
348.8
15. Referring to Scenario 11-1, the within groups degrees of freedom is
a) 3
b) 4
c) 16
d) 19
Analysis of Variance 11-5
16. Referring to Scenario 11-1, the total degrees of freedom is
a) 3
b) 4
c) 16
d) 19
17. Referring to Scenario 11-1, the among-group (between-group) mean squares is
a) 8.525
b) 70.8
c) 212.4
d) 637.2
18. Referring to Scenario 11-1, at a significance level of 1%,
a) there is insufficient evidence to conclude that the mean number of customers bumped by
the 4 packages are not all the same.
b) there is insufficient evidence to conclude that the mean number of customers bumped by
the 4 packages are all the same.
c) there is sufficient evidence to conclude that the mean number of customers bumped by
the 4 packages are not all the same.
d) there is sufficient evidence to conclude that the mean number of customers bumped by
the 4 packages are all the same.
11-6 Analysis of Variance
SCENARIO 11-2
A realtor wants to compare the mean sales–to-appraisal ratios of residential properties sold in four
neighborhoods (A, B, C, and D). Four properties are randomly selected from each neighborhood and
the ratios recorded for each, as shown below.
A: 1.2, 1.1, 0.9, 0.4 C: 1.0, 1.5, 1.1, 1.3
B: 2.5, 2.1, 1.9, 1.6 D: 0.8, 1.3, 1.1, 0.7
Interpret the results of the analysis summarized in the following table:
Source df SS MS F PR > F
Neighborhoods 3.1819 1.0606 10.76 0.001
Error 12
Total 4.3644
19. Referring to Scenario 11-2, the among group degrees of freedom is
a) 3
b) 4
c) 12
d) 16
20. Referring to Scenario 11-2, the within group sum of squares is
a) 1.0606
b) 1.1825
c) 3.1819
d) 4.3644
21. Referring to Scenario 11-2, the within group mean squares is
a) 0.0985
b) 0.2910
c) 1.0606
d) 1.1825
ANSWER:
Analysis of Variance 11-7
22. Referring to Scenario 11-2,
a) at the 0.05 level of significance, the mean ratios for the 4 neighborhoods are not all the
same.
b) at the 0.01 level of significance, the mean ratios for the 4 neighborhoods are all the same.
c) at the 0.10 level of significance, the mean ratios for the 4 neighborhoods are not
significantly different.
d) at the 0.05 level of significance, the mean ratios for the 4 neighborhoods are not
significantly different from 0.
23. Referring to Scenario 11-2, the null hypothesis for Levene’s test for homogeneity of variances is
a)
0:A B C D
H
b)
0:A B C D
H M M M M
c)
2 2 2 2
0:A B C D
H
d)
0:A B C D
H
24. Referring to Scenario 11-2, the value of the test statistic for Levene’s test for homogeneity of
variances is
a) 0.25
b) 0.37
c) 4.36
d) 10.76
11-8 Analysis of Variance
25. Referring to Scenario 11-2, the numerator and denominator degrees of freedom for Levene’s test
for homogeneity of variances at a 5% level of significance are, respectively,
a) 3, 12
b) 12, 3
c) 3, 15
d) 15, 3
26. Referring to Scenario 11-2, the critical value of Levene’s test for homogeneity of variances at a
5% level of significance is
a) 0.64
b) 2.48
c) 3.29
d) 3.49
27. Referring to Scenario 11-2, the p-value of the test statistic for Levene’s test for homogeneity of
variances is
a) 0.25
b) 0.64
c) 0.86
d) 3.49
28. Referring to Scenario 11-2, what should be the decision for the Levene’s test for homogeneity of
variances at a 5% level of significance?
a) Reject the null hypothesis because the p-value is smaller than the level of significance.
b) Reject the null hypothesis because the p-value is larger than the level of significance.
c) Do not reject the null hypothesis because the p-value is smaller than the level of
significance.
d) Do not reject the null hypothesis because the p-value is larger than the level of
significance.
Analysis of Variance 11-9
29. Referring to Scenario 11-2, what should be the conclusion for the Levene’s test for homogeneity
of variances at a 5% level of significance?
a) There is insufficient evidence that the variances are all the same.
b) There is sufficient evidence that the variances are all the same.
c) There is insufficient evidence that the variances are not all the same.
d) There is sufficient evidence that the variances are not all the same.
SCENARIO 11-3
As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting
speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum downhill
speeds. The results are presented in miles per hour in the table below.
Trial Barth Tornado Reiser Shaw
1 43 37 41 43
2 46 38 45 45
3 43 39 42 46
30. Referring to Scenario 11-3, the sporting goods retailer decided to perform an ANOVA F test.
The amount of total variation or SST is __________.
31. Referring to Scenario 11-3, the among group variation or SSA is __________.
32. Referring to Scenario 11-3, the within group variation or SSW is __________.
ANSWER:
11-10 Analysis of Variance
33. Referring to Scenario 11-3, the value of MSA is __________, while MSW is __________.
34. Referring to Scenario 11-3, the null hypothesis that the mean downhill coasting speeds of the 4
brands of bicycles are equal will be rejected at a level of significance of 0.05 if the value of the
test statistic is greater than __________.
35. Referring to Scenario 11-3, in testing the null hypothesis that the mean downhill coasting speeds
of the 4 brands of bicycles are equal, the value of the test statistic is __________.
36. Referring to Scenario 11-3, construct the ANOVA table from the sample data.
37. True or False: Referring to Scenario 11-3, the null hypothesis should be rejected at a 5% level of
significance.
Analysis of Variance 11-11
38. True or False: Referring to Scenario 11-3, the decision made implies that all 4 means are
significantly different.
39. True or False: Referring to Scenario 11-3, the test is valid only if the population of speeds has the
same variance for the 4 brands.
40. True or False: Referring to Scenario 11-3, the test is less sensitive to the assumption that the
population of speeds has the same variance for the 4 brands because the sample sizes of the 4
brands are equal.
41. True or False: Referring to Scenario 11-3, the test is valid only if the population of speeds is
normally distributed.
42. True or False: Referring to Scenario 11-3, the test is robust to the violation of the assumption that
the population of speeds is normally distributed.
43. Referring to Scenario 11-3, the sporting goods retailer decided to compare the 4 treatment means
by using the Tukey-Kramer procedure with an overall level of significance of 0.05. There are
________ pairwise comparisons that can be made.
11-12 Analysis of Variance
44. Referring to Scenario 11-3, using an overall level of significance of 0.05, the critical value of the
Studentized range Q used in calculating the critical range for the Tukey-Kramer procedure is
________.
45. Referring to Scenario 11-3, using an overall level of significance of 0.05, the critical range for
the Tukey-Kramer procedure is ________.
46. True or False: Referring to Scenario 11-3, based on the Tukey-Kramer procedure with an overall
level of significance of 0.05, the retailer would decide that there is a significant difference
between all pairs of mean speeds.
47. True or False: Referring to Scenario 11-3, based on the Tukey-Kramer procedure with an overall
level of significance of 0.05, the retailer would decide that there is no significant difference
between any pair of mean speeds.
48. True or False: Referring to Scenario 11-3, based on the Tukey-Kramer procedure with an overall
level of significance of 0.05, the retailer would decide that the mean speed for the Tornado brand
is significantly different from each of the mean speeds for other brands.
Analysis of Variance 11-13
49. True or False: Referring to Scenario 11-3, based on the Tukey-Kramer procedure with an overall
level of significance of 0.05, the retailer would decide that the 3 means other than the mean for
Tornado are not significantly different from each other.
50. Referring to Scenario 11-3, the null hypothesis for Levene’s test for homogeneity of variances is
a)
0:A B C D
H
b)
0:A B C D
H M M M M
c)
2 2 2 2
0:A B C D
H
d)
0:A B C D
H
51. Referring to Scenario 11-3, what is the value of the test statistic for Levene’s test for
homogeneity of variances?
52. Referring to Scenario 11-3, what are the numerator and denominator degrees of freedom for
Levene’s test for homogeneity of variances respectively?
53. Referring to Scenario 11-3, what is the critical value of Levene’s test for homogeneity of
variances at a 5% level of significance?
11-14 Analysis of Variance
54. Referring to Scenario 11-3, what is the p-value of the test statistic for Levene’s test for
homogeneity of variances?
55. Referring to Scenario 11-3, what should be the decision for the Levene’s test for homogeneity of
variances at a 5% level of significance?
a) Reject the null hypothesis because the p-value is smaller than the level of significance.
b) Reject the null hypothesis because the p-value is larger than the level of significance.
c) Do not reject the null hypothesis because the p-value is smaller than the level of
significance.
d) Do not reject the null hypothesis because the p-value is larger than the level of
significance.
56. Referring to Scenario 11-3, what should be the conclusion for the Levene’s test for homogeneity
of variances at a 5% level of significance?
a) There is insufficient evidence that the variances are all the same.
b) There is sufficient evidence that the variances are all the same.
c) There is insufficient evidence that the variances are not all the same.
d) There is sufficient evidence that the variances are not all the same.
SCENARIO 11-4
An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants 15 fields,
5 with each variety. She then measures the crop yield in bushels per acre. Treating this as a
completely randomized design, the results are presented in the table that follows.
Trial Smith Walsh Trevor
1 11.1 19.0 14.6
2 13.5 18.0 15.7
3 15.3 19.8 16.8
4 14.6 19.6 16.7
5 9.8 16.6 15.2
Analysis of Variance 11-15
57. Referring to Scenario 11-4, the agronomist decided to perform an ANOVA F test. The amount
of total variation or SST is __________.
58. Referring to Scenario 11-4, the among-group variation or SSA is __________.
59. Referring to Scenario 11-4, the within-group variation or SSW is __________.
60. Referring to Scenario 11-4, the value of MSA is __________, while MSW is __________.
61. Referring to Scenario 11-4, the null hypothesis will be rejected at a level of significance of 0.01
if the value of the test statistic is greater than __________.
62. Referring to Scenario 11-4, the value of the test statistic is __________.
11-16 Analysis of Variance
63. Referring to Scenario 11-4, construct the ANOVA table from the sample data.
64. Referring to Scenario 11-4, state the null hypothesis that can be tested.
65. True or False: Referring to Scenario 11-4, the null hypothesis should be rejected at 0.005 level of
significance.
66. True or False: Referring to Scenario 11-4, the decision made at 0.005 level of significance
implies that all 3 means are significantly different.
67. True or False: Referring to Scenario 11-4, the test is valid only if the population of crop yields
has the same variance for the 3 varieties.
Analysis of Variance 11-17
68. True or False: Referring to Scenario 11-4, the test is valid only if the population of crop yields is
normally distributed for the 3 varieties.
69. Referring to Scenario 11-4, the agronomist decided to compare the 3 treatment means by using
the Tukey-Kramer procedure with an overall level of significance of 0.01. There are ________
pairwise comparisons that can be made.
70. Referring to Scenario 11-4, using an overall level of significance of 0.01, the critical value of the
Studentized range Q used in calculating the critical range for the Tukey-Kramer procedure is
________.
71. Referring to Scenario 11-4, using an overall level of significance of 0.01, the critical range for
the Tukey-Kramer procedure is ________.
72. True or False: Referring to Scenario 11-4, based on the Tukey-Kramer procedure with an overall
level of significance of 0.01, the agronomist would decide that there is a significant difference
between the crop yield of Smith and Walsh seeds.
ANSWER:
11-18 Analysis of Variance
73. True or False: Referring to Scenario 11-4, based on the Tukey-Kramer procedure with an overall
level of significance of 0.01, the agronomist would decide that there is a significant difference
between the crop yield of Smith and Trevor seeds.
74. True or False: Referring to Scenario 11-4, based on the Tukey-Kramer procedure with an overall
level of significance of 0.01, the agronomist would decide that there is a significant difference
between the crop yield of Walsh and Trevor seeds.
SCENARIO 11-5
A hotel chain has identically small sized resorts in 5 locations in different small islands. The data
that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5
locations.
ROW Location A Location B Location C Location D Location E
1 28 40 21 37 22
2 33 35 21 47 19
3 41 33 27 45 25
Analysis of Variance
Source df SS MS F p
Location 4 963.6 11.47 0.001
Error 10 210.0
Total
75. True or False: Referring to Scenario 11-5, if a level of significance of 0.05 is chosen, the null
hypothesis should be rejected.
ANSWER:
76. True or False: Referring to Scenario 11-5, if a level of significance of 0.05 is chosen, the
decision made indicates that all 5 locations have different mean occupancy rates.
Analysis of Variance 11-19
77. True or False: Referring to Scenario 11-5, if a level of significance of 0.05 is chosen, the
decision made indicates that at least 2 of the 5 locations have different mean occupancy rates.
78. Referring to Scenario 11-5, the among-group variation or SSA is _________.
79. Referring to Scenario 11-5, the within-group variation or SSW is _________.
80. Referring to Scenario 11-5, the total variation or SST is ________.
81. Referring to Scenario 11-5, the value of MSA is ______ while MSW is _______.
82. Referring to Scenario 11-5, the numerator and denominator degrees of freedom of the test ratio
are ________ and ________, respectively.
83. True or False: Referring to Scenario 11-5, the total mean squares is 261.90.
ANSWER:
11-20 Analysis of Variance
84. Referring to Scenario 11-5, the null hypothesis for Levene’s test for homogeneity of variances is
a)
0:A B C D
H
b)
0:A B C D
H M M M M
c)
2 2 2 2
0:A B C D
H
d)
0:A B C D
H
85. Referring to Scenario 11-5, what is the value of the test statistic for Levene’s test for
homogeneity of variances?
86. Referring to Scenario 11-5, what are the numerator and denominator degrees of freedom for
Levene’s test for homogeneity of variances respectively?
87. Referring to Scenario 11-5, what is the critical value of Levene’s test for homogeneity of
variances at a 5% level of significance?
88. Referring to Scenario 11-5, what is the p-value of the test statistic for Levene’s test for
homogeneity of variances?
Analysis of Variance 11-21
89. Referring to Scenario 11-5, what should be the decision for the Levene’s test for homogeneity of
variances at a 5% level of significance?
a) Reject the null hypothesis because the p-value is smaller than the level of significance.
b) Reject the null hypothesis because the p-value is larger than the level of significance.
c) Do not reject the null hypothesis because the p-value is smaller than the level of
significance.
d) Do not reject the null hypothesis because the p-value is larger than the level of
significance.
90. Referring to Scenario 11-5, what should be the conclusion for the Levene’s test for homogeneity
of variances at a 5% level of significance?
a) There is insufficient evidence that the variances are all the same.
b) There is sufficient evidence that the variances are all the same.
c) There is insufficient evidence that the variances are not all the same.
d) There is sufficient evidence that the variances are not all the same.
91. True or False: The F test in a completely randomized model is just an expansion of the t test for
independent samples.
92. True or False: A completely randomized design with 4 groups would have 6 possible pairwise
comparisons.
11-22 Analysis of Variance
93. Interaction in an experimental design can be tested in
a) a completely randomized model.
b) a two-factor model.
c) a Tukey-Kramer procedure
d) all ANOVA models.
94. In a two-way ANOVA, the degrees of freedom for the interaction term is
a) (r – 1)(c – 1).
b) rc(n – 1).
c) (r – 1).
d) rcn + 1.
95. In a two-way ANOVA, the degrees of freedom for the “error” term is
a) (r – 1)(c – 1).
b) rc(n’ – 1).
c) (r – 1).
d) rcn’ + 1.
96. True or False: In a two-factor ANOVA analysis, the sum of squares due to both factors, the
interaction sum of squares and the within sum of squares must add up to the total sum of squares.
97. True or False: In a two-way ANOVA, it is appropriate to interpret main effects when the
interaction component is not significant.
Analysis of Variance 11-23
SCENARIO 11-6
An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3
varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in
bushels per acre. Treating this as a randomized block design, the results are presented in the table
that follows.
Fields Smith Walsh Trevor
1 11.1 19.0 14.6
2 13.5 18.0 15.7
3 15.3 19.8 16.8
4 14.6 19.6 16.7
5 9.8 16.6 15.2
98. Referring to Scenario 11-6, the agronomist decided to perform a randomized block F test for the
difference in the means. The amount of total variation or SST is __________.
99. Referring to Scenario 11-6, the among-group variation or SSA is __________.
100. Referring to Scenario 11-6, the among-block variation or SSBL is __________.
KEYWORDS: randomized block design, sum of squares
101. Referring to Scenario 11-6, the value of MSA is __________, while MSBL is __________.
11-24 Analysis of Variance
102. Referring to Scenario 11-6, the null hypothesis for the randomized block F test for the
difference in the means is
a)
0 Field 1 Field 2 Field 3 Field 4 Field 5
:H
b)
0 Smith Walsh Trevor
:H
c)
0 Field 1 Field 2 Field 3 Field 4 Field 5
:H M M M M M
d)
0 Smith Walsh Trevor
:H M M M
103. Referring to Scenario 11-6, what are the degrees of freedom of the randomized block F test for
the difference in the means at a level of significance of 0.01?
104. Referring to Scenario 11-6, what is the critical value of the randomized block F test for the
difference in the means at a level of significance of 0.01?
105. Referring to Scenario 11-6, what is the value of the test statistic for the randomized block F
test for the difference in the means?
106. Referring to Scenario 11-6, what is the p-value of the test statistic for the randomized block F
test for the difference in the means?
Analysis of Variance 11-25
107. True or False: Referring to Scenario 11-6, the null hypothesis for the randomized block F test
for the difference in the means should be rejected at a 0.01 level of significance.
108. True or False: Referring to Scenario 11-6, the decision made at a 0.01 level of significance on
the randomized block F test for the difference in means implies that all 3 means are significantly
different.
109. True or False: Referring to Scenario 11-6, the randomized block F test is valid only if the
population of crop yields has the same variance for the 3 varieties.
110. True or False: Referring to Scenario 11-6, the randomized block F test is valid only if the
population of crop yields is normally distributed for the 3 varieties.
111. True or False: Referring to Scenario 11-6, the randomized block F test is valid only if there is
no interaction between the variety of seeds and the patches of fields.
112. Referring to Scenario 11-6, the agronomist decided to compare the 3 treatment means by using
the Tukey multiple comparison procedure with an overall level of significance of 0.01. How
many pairwise comparisons can be made?
11-26 Analysis of Variance
113. Referring to Scenario 11-6, using an overall level of significance of 0.01, what is the critical
value of the Studentized range Q used in calculating the critical range for the Tukey multiple
comparison procedure?
114. Referring to Scenario 11-6, using an overall level of significance of 0.01, what is the critical
range for the Tukey multiple comparison procedure?
115. True or False: Referring to Scenario 11-6, based on the Tukey multiple comparison procedure
with an overall level of significance of 0.01, the agronomist would decide that there is a
significant difference between the crop yield of Smith and Walsh seeds.
116. True or False: Referring to Scenario 11-6, based on the Tukey-Kramer procedure with an
overall level of significance of 0.01, the agronomist would decide that there is a significant
difference between the crop yield of Smith and Trevor seeds.
117. True or False: Referring to Scenario 11-6, based on the Tukey multiple comparison procedure
with an overall level of significance of 0.01, the agronomist would decide that there is a
significant difference between the crop yield of Walsh and Trevor seeds.
ANSWER:
Analysis of Variance 11-27
118. Referring to Scenario 11-6, what is the null hypothesis for testing the block effects?
a)
0 Field 1 Field 2 Field 3 Field 4 Field 5
:H
b)
0 Smith W alsh Trevor
:H
c)
0 Field 1 Field 2 Field 3 Field 4 Field 5
:H M M M M M
d)
0 Smith W alsh Trevor
:H M M M
119. Referring to Scenario 11-6, what are the degrees of freedom of the F test statistic for testing the
block effects?
120. Referring to Scenario 11-6, what is the value of the F test statistic for testing the block effects?
121. Referring to Scenario 11-6, what is the critical value for testing the block effects at a 0.01 level
of significance?
122. Referring to Scenario 11-6, what is the p-value of the F test statistic for testing the block
effects?
11-28 Analysis of Variance
123. True or False: Referring to Scenario 11-6, the null hypothesis for the F test for the block
effects should be rejected at a 0.01 level of significance.
124. True or False: Referring to Scenario 11-6, the decision made at a 0.01 level of significance on
the F test for the block effects implies that the blocking has been advantageous in reducing the
experiment error.
125. Referring to Scenario 11-6, what is the estimated relative efficiency?
126. True or False: Referring to Scenario 11-6, the relative efficiency means that 2.47 times as
many observations in each variety group would be needed in a one–way ANOVA design as
compared to the randomized block design in order to obtain the same precision for comparison of
the variety means.
SCENARIO 11-7
A student team in a business statistics course designed an experiment to investigate whether the
brand of bubblegum used affected the size of bubbles they could blow. To reduce the person–to–
person variability, the students decided to use a randomized block design using themselves as blocks.
Four brands of bubblegum were tested. A student chewed two pieces of a brand of gum and then
blew a bubble, attempting to make it as big as possible. Another student measured the diameter of the
bubble at its biggest point. The following table gives the diameters of the bubbles (in inches) for the
16 observations.
Brand of Bubblegum
Student
A
B
C
D
Kyle
8.75
9.50
8.50
11.50
Sarah
9.50
4.00
8.50
11.00
Leigh
9.25
5.50
7.50
7.50
Isaac
9.50
8.50
7.50
7.50
Analysis of Variance 11-29
127. Referring to Scenario 11-7, the amount of total variation or SST is __________.
KEYWORDS: randomized block design, sum of squares
128. Referring to Scenario 11-7, the among-group variation or SSA is __________.
129. Referring to Scenario 11-7, the among-block variation or SSBL is __________.
130. Referring to Scenario 11-7, the value of MSA is __________, while MSBL is __________.
131. Referring to Scenario 11-7, the null hypothesis for the randomized block F test for the
difference in the means is
a)
DCBA
H
:
0
b)
IssacLeighSarahKyle
H
:
0
c)
DCBA MMMMH :
0
d)
IssacLeighSarahKyle MMMMH :
0
132. Referring to Scenario 11-7, what are the degrees of freedom of the randomized block F test for
the difference in the means at a level of significance of 0.01?
11-30 Analysis of Variance
133. Referring to Scenario 11-7, what is the critical value of the randomized block F test for the
difference in the means at a level of significance of 0.05?
134. Referring to Scenario 11-7, what is the value of the test statistic for the randomized block F
test for the difference in the means?
135. Referring to Scenario 11-7, what is the p-value of the test statistic for the randomized block F
test for the difference in the means?
136. True or False: Referring to Scenario 11-7, the null hypothesis for the randomized block F test
for the difference in the means should be rejected at a 0.05 level of significance.
137. True or False: Referring to Scenario 11-7, the decision made at a 0.05 level of significance on
the randomized block F test for the difference in means implies that all 4 means are significantly
different.
138. True or False: Referring to Scenario 11-7, the randomized block F test is valid only if the
population of diameters has the same variance for the 4 brands.
ANSWER:
Analysis of Variance 11-31
139. True or False: Referring to Scenario 11-7, the randomized block F test is valid only if the
population of diameters is normally distributed for the 4 brands.
140. True or False: Referring to Scenario 11-7, the randomized block F test is valid only if there is
no interaction between the diameters of the 4 brands of bubble gums and the 4 ability of the
students.
141. Referring to Scenario 11-7, is it appropriate to use the Tukey multiple comparison procedure
based on the test result above?
142. Referring to Scenario 11-7, what is the null hypothesis for testing the block effects?
a)
DCBA
H
:
0
b)
IssacLeighSarahKyle
H
:
0
c)
DCBA MMMMH :
0
d)
IssacLeighSarahKyle MMMMH :
0
143. Referring to Scenario 11-7, what are the degrees of freedom of the F test statistic for testing the
block effects?
11-32 Analysis of Variance
144. Referring to Scenario 11-7, what is the value of the F test statistic for testing the block effects?
145. Referring to Scenario 11-7, what is the critical value for testing the block effects at a 0.05 level
of significance?
146. Referring to Scenario 11-7, what is the p-value of the F test statistic for testing the block
effects?
147. True or False: Referring to Scenario 11-7, the null hypothesis for the F test for the block
effects should be rejected at a 0.05 level of significance.
148. True or False: Referring to Scenario 11-7, the decision made at a 0.05 level of significance on
the F test for the block effects implies that the blocking has been advantageous in reducing the
experiment error.
149. Referring to Scenario 11-7, what is the estimated relative efficiency?
Analysis of Variance 11-33
150. True or False: Referring to Scenario 11-7, the relative efficiency means that 1.0144 times as
many observations in each brand would be needed in a one-way ANOVA design as compared to
the randomized block design in order to obtain the same precision for comparison of the different
means.
SCENARIO 11-8
An important factor in selecting database software is the time required for a user to learn how to use
the system. To evaluate three potential brands (A, B and C) of database software, a company
designed a test involving five different employees. To reduce variability due to differences among
employees, each of the five employees is trained on each of the three different brands. The amount
of time (in hours) needed to learn each of the three different brands is given below:
Software
Operator
A
B
C
1
17
17
23
2
18
17
23
3
14
13
19
4
12
11
18
5
19
17
22
Mean
16
15
21
Below is the Excel output for the randomized block design:
Source of Variation
SS
df
MS
F
P-value
F crit
Rows
84.66667
4
21.16667
50.8
9.98E-06
3.837853
Columns
103.3333
2
51.66667
124
9.54E-07
4.45897
Error
3.333333
8
0.416667
Total
191.3333
14
151. Referring to Scenario 11-8, the amount of total variation or SST is __________.
152. Referring to Scenario 11-8, the among-group variation or SSA is __________.
11-34 Analysis of Variance
153. Referring to Scenario 11-8, the among-block variation or SSBL is __________.
154. Referring to Scenario 11-8, the value of MSA is __________, while MSBL is __________.
155. Referring to Scenario 11-8, the null hypothesis for the randomized block F test for the
difference in the means is
a)
0:A B C
H
b)
0 1 2 3 4 5
:H
c)
0:A B C
H M M M
d)
0 1 2 3 4 5
:H M M M M M
156. Referring to Scenario 11-8, what are the degrees of freedom of the randomized block F test for
the difference in the means at a level of significance of 0.05?
157. Referring to Scenario 11-8, what is the critical value of the randomized block F test for the
difference in the means at a level of significance of 0.05?
Analysis of Variance 11-35
158. Referring to Scenario 11-8, what is the value of the test statistic for the randomized block F
test for the difference in the means?
159. Referring to Scenario 11-8, what is the p-value of the test statistic for the randomized block F
test for the difference in the means?
160. True or False: Referring to Scenario 11-8, the null hypothesis for the randomized block F test
for the difference in the means should be rejected at a 0.05 level of significance.
161. True or False: Referring to Scenario 11-8, the decision made at a 0.05 level of significance on
the randomized block F test for the difference in means implies that all 3 means are significantly
different from each other.
162. True or False: Referring to Scenario 11-8, the randomized block F test is valid only if the
population of the amount of time needed has the same variance for the 3 brands.
163. True or False: Referring to Scenario 11-8, the randomized block F test is valid only if the
population of the amount of time needed is normally distributed for the 3 brands.
11-36 Analysis of Variance
164. True or False: Referring to Scenario 11-8, the randomized block F test is valid only if there is
no interaction between the amount of time needed on the 3 brands of software and the 5
employees.
165. True or False: Referring to Scenario 11-8, it is appropriate to use the Tukey multiple
comparison procedure based on the test result above.
166. Referring to Scenario 11-8, what are the degrees of freedom of the Studentized range
distribution for the Tukey multiple comparison procedure?
167. Referring to Scenario 11-8, what is the upper-tail critical value of the Studentized range
distribution for the Tukey multiple comparison procedure at the 5% level of significance?
168. Referring to Scenario 11-8, what is the critical range of the Studentized range distribution for
the Tukey multiple comparison procedure at the 5% level of significance?
169. True or False: Referring to Scenario 11-8, there is evidence of a significant difference in the
mean amount of time needed to learn Brand A and Brand B at the 5% level of significance?
Analysis of Variance 11-37
170. True or False: Referring to Scenario 11-8, there is evidence of a significant difference in the
mean amount of time needed to learn Brand A and Brand C at the 5% level of significance?
171. True or False: Referring to Scenario 11-8, there is evidence of a significant difference in the
mean amount of time needed to learn Brand B and Brand C at the 5% level of significance?
172. Referring to Scenario 11-8, what is the null hypothesis for testing the block effects?
a)
0:A B C
H
b)
0 1 2 3 4 5
:H
c)
0:A B C
H M M M
d)
0 1 2 3 4 5
:H M M M M M
173. Referring to Scenario 11-8, what are the degrees of freedom of the F test statistic for testing the
block effects?
174. Referring to Scenario 11-8, what is the value of the F test statistic for testing the block effects?
11-38 Analysis of Variance
175. Referring to Scenario 11-8, what is the critical value for testing the block effects at a 0.05 level
of significance?
176. Referring to Scenario 11-8, what is the p-value of the F test statistic for testing the block
effects?
177. True or False: Referring to Scenario 11-8, the null hypothesis for the F test for the block
effects should be rejected at a 0.05 level of significance.
178. True or False: Referring to Scenario 11-8, the decision made at a 0.05 level of significance on
the F test for the block effects implies that the blocking has been advantageous in reducing the
experiment error.
179. Referring to Scenario 11-8, what is the estimated relative efficiency?
180. True or False: Referring to Scenario 11-8, the relative efficiency means that 15.23 times as
many observations in each brand would be needed in a one-way ANOVA design as compared to
the randomized block design in order to obtain the same precision for comparison of the different
means.
Analysis of Variance 11-39
SCENARIO 11-9
Psychologists have found that people are generally reluctant to transmit bad news to their peers. This
phenomenon has been termed the “MUM effect.” To investigate the cause of the MUM effect, 40
undergraduates at Duke University participated in an experiment. Each subject was asked to
administer an IQ test to another student and then provide the test taker with his or her percentile
score. Unknown to the subject, the test taker was a bogus student who was working with the
researchers. The experimenters manipulated two factors: subject visibility and success of test taker,
each at two levels. Subject visibility was either visible or not visible to the test taker. Success of the
test taker was either top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2
x 2 = 4 experimental conditions, then the time (in seconds) between the end of the test and the
delivery of the percentile score from the subject to the test taker was measured. (This variable is
called the latency to feedback.) The data were subjected to appropriate analyses with the following
results.
Source df SS MS F PR > F
Subject visibility 1 1380.24 1380.24 4.26 0.043
Test taker success 1 1325.16 1325.16 4.09 0.050
Interaction 1 3385.80 3385.80 10.45 0.002
Error 36 11,664.00 324.00
Total 39 17,755.20
181. Referring to Scenario 11-9, what type of experimental design was employed in this study?
a) Completely randomized design with 4 treatments
b) Randomized block design with four treatments and 10 blocks
c) 2 x 2 factorial design with 10 observations
d) None of the above
182. Referring to Scenario 11-9, at the 0.01 level, what conclusions can you reach from the
analysis?
a) At the 0.01 level, subject visibility and test taker success are significant predictors of
latency feedback.
b) At the 0.01 level, the model is not useful for predicting latency to feedback.
c) At the 0.01 level, there is evidence to indicate that subject visibility and test taker
success interact.
d) At the 0.01 level, there is no evidence of interaction between subject visibility and test
taker success.
11-40 Analysis of Variance
183. Referring to Scenario 11-9, in the context of this study, interpret the statement: “Subject
visibility and test taker success interact.”
a) The difference between the mean feedback time for visible and nonvisible subjects
depends on the success of the test taker.
b) The difference between the mean feedback time for test takers scoring in the top 20%
and bottom 20% depends on the visibility of the subject.
c) The relationship between feedback time and subject visibility depends on the success of
the test taker.
d) All of the above are correct interpretations.
complex, multilevel building on campus. Specifically, he wanted to determine whether different
building signs (building maps versus wall signage) affect the total amount of time visitors require
to reach their destination and whether that time depends on whether the starting location is inside
or outside the building. Three subjects were assigned to each of the combinations of signs and
starting locations, and travel time in seconds from beginning to destination was recorded. How
should the data be analyzed?
Starting Room
Interior Exterior
Wall Signs 141, 119, 238 224, 339, 139
Map 85, 94, 126 226, 129, 130
a) Completely randomized design
b) Randomized block design
c) 2 x 2 factorial design
d) Levene’s test
Analysis of Variance 11-41
SCENARIO 11-10
A campus researcher wanted to investigate the factors that affect visitor travel time in a complex,
multilevel building on campus. Specifically, he wanted to determine whether different building signs
(building maps versus wall signage) affect the total amount of time visitors require to reach their
destination and whether that time depends on whether the starting location is inside or outside the
building. Three subjects were assigned to each of the combinations of signs and starting locations,
and travel time in seconds from beginning to destination was recorded. An Excel output of the
appropriate analysis is given below:
ANOVA
Source of Variation
SS
df
MS
F
P-value
F crit
Signs
14008.33
14008.33
0.11267
5.317645
Starting Location
12288
2.784395
0.13374
5.317645
Interaction
48
48
0.919506
5.317645
Within
35305.33
4413.167
Total
61649.67
11
185. Referring to Scenario 11-10, the degrees of freedom for the different building signs (factor A)
is a) 1
b) 2
c) 3
d) 8
186. Referring to Scenario 11-10, the within (error) degrees of freedom is
a) 1
b) 4
c) 8
d) 11
11-42 Analysis of Variance
187. Referring to Scenario 11-10, the mean squares for starting location (factor B) is
a) 48
b) 4,413.17
c) 12,288
d) 14,008.3
188. Referring to Scenario 11-10, the F test statistic for testing the main effect of types of signs is
a) 0.0109
b) 2.7844
c) 3.1742
d) 5.3176
189. Referring to Scenario 11-10, the F test statistic for testing the interaction effect between the
types of signs and the starting location is
a) 0.0109
b) 2.7844
c) 3.1742
d) 5.3176
190. Referring to Scenario 11-10, at 1% level of significance,
a) there is insufficient evidence to conclude that the difference between the mean traveling
time for the different starting locations depends on the types of signs.
b) there is insufficient evidence to conclude that the difference between the mean traveling
time for the different types of signs depends on the starting locations.
c) there is insufficient evidence to conclude that the relationship between traveling time and
the types of signs depends on the starting locations.
d) All of the above.
Analysis of Variance 11-43
191. Referring to Scenario 11-10, at 10% level of significance,
a) there is sufficient evidence to conclude that the difference between the mean traveling
time for the different starting locations depends on the types of signs.
b) there is insufficient evidence to conclude that the difference between the mean traveling
time for the different types of signs depends on the starting locations.
c) there is sufficient evidence to conclude that the difference between the mean traveling
time for the different starting locations does not depend on the types of signs.
d) None of the above.
SCENARIO 11-11
A physician and president of a Tampa Health Maintenance Organization (HMO) are attempting to
show the benefits of managed health care to an insurance company. The physician believes that
certain types of doctors are more cost-effective than others. One theory is that Primary Specialty is an
important factor in measuring the cost-effectiveness of physicians. To investigate this, the president
obtained independent random samples of 20 HMO physicians from each of 4 primary specialties –
General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physicians (FP) – and
recorded the total charges per member per month for each. A second factor which the president
believes influences total charges per member per month is whether the doctor is a foreign or USA
medical school graduate. The president theorizes that foreign graduates will have higher mean
charges than USA graduates. To investigate this, the president also collected data on 20 foreign
medical school graduates in each of the 4 primary specialty types described above. So information on
charges for 40 doctors (20 foreign and 20 USA medical school graduates) was obtained for each of
the 4 specialties. The results for the ANOVA are summarized in the following table.
Source df SS MS F PR > F
Specialty 3 22,855 7,618 60.94 0.0001
Med school 1 105 105 0.84 0.6744
Interaction 3 890 297 2.38 0.1348
Error 152 18,950
Total 159 42,800
192. Referring to Scenario 11-11, what was the total number of doctors included in the study?
a) 20
b) 40
c) 159
d) 160
11-44 Analysis of Variance
193. Referring to Scenario 11-11, what degrees of freedom should be used to determine the critical
value of the F ratio against which to test for interaction between the two factors?
a) numerator df = 1, denominator df = 159
b) numerator df = 3, denominator df = 159
c) numerator df = 1, denominator df = 152
d) numerator df = 3, denominator df = 152
194. Referring to Scenario 11-11, interpret the test for interaction.
a) There is insufficient evidence to say at the 0.10 level of significance that the difference
between the mean charges for foreign and USA graduates depends on primary specialty.
b) There is sufficient evidence to say at the 0.10 level of significance that the difference
between the mean charges for foreign and USA graduates depends on primary specialty.
c) There is sufficient evidence at the 0.10 level of significance of a difference between the
mean charges for foreign and USA medical graduates.
d) There is sufficient evidence to say at the 0.10 level of significance that mean charges
depend on both primary specialty and medical school.
195. Referring to Scenario 11-11, what degrees of freedom should be used to determine the critical
value of the F ratio against which to test for differences in the mean charges for doctors among
the four primary specialty areas?
a) numerator df = 1, denominator df = 159
b) numerator df = 3, denominator df = 159
c) numerator df = 1, denominator df = 152
d) numerator df = 3, denominator df = 152
Analysis of Variance 11-45
196. Referring to Scenario 11-11, what degrees of freedom should be used to determine the critical
value of the F ratio against which to test for differences between the mean charges of foreign and
USA medical school graduates?
a) numerator df = 1, denominator df = 159
b) numerator df = 3, denominator df = 159
c) numerator df = 1, denominator df = 152
d) numerator df = 3, denominator df = 152
197. Referring to Scenario 11-11, is there evidence of a difference between the mean charges of
foreign and USA medical school graduates?
a) Yes, the test for the main effect for primary specialty is significant at
= 0.10.
b) No, the test for the main effect for medical school is not significant at
= 0.10.
c) No, the test for the interaction is not significant at
= 0.10.
d) Maybe, but we need information on the
-estimates to fully answer the question.
198. Referring to Scenario 11-11, what assumption(s) need(s) to be made in order to conduct the
test for differences between the mean charges of foreign and USA medical school graduates?
a) There is no significant interaction effect between the area of primary specialty and the
medical school on the doctors’ mean charges.
b) The charges in each group of doctors sampled are drawn from normally distributed
populations.
c) The charges in each group of doctors sampled are drawn from populations with equal
variances.
d) All of the above are necessary assumptions.
11-46 Analysis of Variance
SCENARIO 11-12
The marketing manager of a company producing a new cereal aimed for children wants to examine
the effect of the color and shape of the box’s logo on the approval rating of the cereal. He combined 4
colors and 3 shapes to produce a total of 12 designs. Each logo was presented to 2 different groups (a
total of 24 groups) and the approval rating for each was recorded and is shown below. The manager
analyzed these data using the
= 0.05 level of significance for all inferences.
COLORS
SHAPES
Red
Green
Blue
Yellow
Circle
54
67
36
45
44
61
44
41
Square
34
56
36
21
36
58
30
25
Diamond
46
60
34
31
48
60
38
33
Analysis of Variance
Source df SS MS F p
Colors 3 2711.17 903.72 72.30 0.000
Shapes 2 579.00 289.50 23.16 0.000
Interaction 6 150.33 25.06 2.00 0.144
Error 12 150.00 12.50
Total 23 3590.50
199. Referring to Scenario 11-12, the mean square for the factor color is ________.
200. Referring to Scenario 11-12, the mean square for the factor shape is ________.
201. Referring to Scenario 11-12, the mean square for the interaction of color and shape is
________.
Analysis of Variance 11-47
202. Referring to Scenario 11-12, the mean square for error is ________.
203. Referring to Scenario 11-12, the critical value of the test for significant differences between
colors is ________.
204. Referring to Scenario 11-12, the value of the statistic used to test for significant differences
between colors is ________.
205. True or False: Referring to Scenario 11-12, based on the results of the hypothesis test, it
appears that there is a significant effect on the approval rating associated with the color of the
logo.
206. Referring to Scenario 11-12, the critical value in the test for significant differences between
shapes is ________.
207. Referring to Scenario 11-12, the value of the statistic used to test for significant differences
between shapes is ________.
ANSWER:
23.16
TYPE: FI DIFFICULTY: Easy
KEYWORDS: two-factor analysis of variance, F test for factor, test statistic
11-48 Analysis of Variance
208. True or False: Referring to Scenario 11-12, based on the results of the hypothesis test, it
appears that there is a significant effect associated with the shape of the logo.
209. Referring to Scenario 11-12, the critical value in the test for a significant interaction is
________.
210. Referring to Scenario 11-12, the value of the statistic used to test for an interaction is
________.
211. True or False: Referring to Scenario 11-12, based on the results of the hypothesis test, it
appears that there is a significant interaction.
212. Which of the following is an assumption required by the Analysis of Means (ANOM)?
a) The variance of the groups is different.
b) The observations from each of the groups are assumed to be approximately normally
distributed.
c) The shape of the distribution of the observations from the groups is different.
d) The number of observations in each group has to be at least 30.
Analysis of Variance 11-49
213. True or False: The Analysis of Means (ANOM) is more appropriate if you want to identify the
group(s) that has(have) a higher or lower mean than the overall mean while the One–Way
Analysis of Variance (One-way ANOVA) is more appropriate when you just want to see if the
means of all the different groups are the same.