Chapter 12
Analysis of Variance
True/False
1. The F distribution's curve is positively skewed.
Answer: True
2. The test statistic used in ANOVA is Student's t.
Answer: False
3. There is one, unique F distribution for a F-statistic with 29 degrees of freedom in the numerator and 28
degrees of freedom in the denominator.
Answer: True
4. One characteristic of the F distribution is that F cannot be negative.
Answer: True
5. One characteristic of the F distribution is that the computed F can only range between -1 and +1.
Answer: False
6. The F distribution is positively skewed and its values may range from 0 to plus infinity.
Answer: True
7. The shape of the F distribution is determined by the degrees of freedom for the F-statistic, one for the
numerator and one for the denominator.
Answer: True
8. Unlike Student's t distribution, there is only one F distribution.
Answer: False
9. Like Student's t distribution, a change in the degrees of freedom causes a change in the shape of the F
distribution.
Answer: True
10. If the computed value of F is 0.99 and the critical value is 3.89, we would not reject the null
hypothesis.
Answer: True
11. For the hypothesis test, H o : σ 21  σ 2 2 , with n1 = 10 and n2 = 10, the F-test statistic is 2.56. At the
0.01 level of significance, we would reject the null hypothesis.
Answer: False
12. For the hypothesis test, H o : σ 21  σ 2 2 , with n1 = 4 and n2 = 4, the F-test statistic is 50.01. At the
0.01 level of significance, we would reject the null hypothesis.
Answer: True
13. For the hypothesis test, H o : σ 21  σ 2 2 , with n1 = 7 and n2 = 7, the F-test statistic is 2.56. At the
0.05 level of significance, we would reject the null hypothesis.
Answer: False
14. For the hypothesis test, H o : σ 21  σ 2 2 , with n1 = 9 and n2 = 9, the F-test statistic is 4.53. At the
0.05 level of significance, we would reject the null hypothesis.
Answer: True
15. To employ ANOVA, the populations being studied must be approximately normally distributed.
Answer: True
16. To employ ANOVA, the populations should have approximately equal standard deviations.
Answer: True
17. In an ANOVA table, k represents the total number of sample observations and n represents the total
number of treatments.
Answer: False
18. In an ANOVA table, k represents the number of treatments, b represents the number of blocks, and n
represents the total number of sample observations.
Answer: True
19. The alternate hypothesis used in ANOVA is 1   2   3 .
Answer: False
20. The alternate hypothesis for ANOVA states that not all the means are equal.
Answer: True
21. For an ANOVA test, rejection of the null hypothesis does not identify which treatment means differ
significantly.
Answer: True
22. If the computed value of F is 4.01 and the critical value is 2.67, we would conclude that all the
population means are equal.
Answer: False
23. If the computed value of F is 11.1 and the 0.05 level is used, we would assume that a mistake in
arithmetic has been made.
Answer: False
24. If we want to determine which treatment means differ, we compute a confidence interval for the
difference between each pair of means.
Answer: True
25. If a confidence interval for the difference between a pair of treatment means includes 0, then there is
no difference in the pair of treatment means.
Answer: True
26. If the endpoints of a confidence interval for the difference between a pair of treatment means are both
positive numbers, then the treatment means are not different.
Answer: False
27. A treatment is a specific source of variation in a set of data.
Answer: True
28. A blocking effect is a specific source of variation in a set of data.
Answer: True
29. When a blocking effect is included in an ANOVA, the result is a smaller error sum of squares.
Answer: True
30. When a blocking effect is included in an ANOVA, two sources of variation are reported: treatment
variation and block variation.
Answer: False
31. When a blocking effect is included in an ANOVA, the analysis is more likely to detect differences in
the treatment means.
Answer: True
32. The F-statistic to test for a blocking effect is computed as the ratio of the Treatment Mean Square and
the Block Mean Square.
Answer: False
33. In a two-way ANOVA, the sum of the treatment, block, and error degrees of freedom equal the total
degrees of freedom.
Answer: True
34. In a two-way ANOVA, the sum of the treatment and block mean squares equals the error mean
square.
Answer: False
35. In a two-way ANOVA, the sum of the treatment, block, and error sum of squares equals the total sum
of squares.
Answer: True
36. In a two-way ANOVA with interaction, there are two factor effects and an interaction effect.
Answer: True
37. In a two-way ANOVA with treatment and block effects, an interaction effect is also included.
Answer: False
38. In an interaction plot, parallel lines are an indication that there is no interaction effect.
Answer: True
39. Interaction between two factors occurs when the effect of one factor on the response variable is the
same for any value of another factor.
Answer: False
Multiple Choice
40. An F statistic is:
A) a ratio of two means.
B) a ratio of two variances.
C) the difference between three means.
D) a population parameter.
Answer: B
41. What distribution does the F distribution approach as the sample size increases?
A) Binomial
B) Normal
C) Poisson
D) Exponential
Answer: B
42. Which statement is correct about the F distribution?
A) Cannot be negative
B) Cannot be positive
C) Is the same as the t distribution
D) Is the same as the z distribution
Answer: A
43. Analysis of variance is used to
A) compare nominal data.
B) compute t test.
C) compare population proportion.
D) simultaneously compare several population means.
Answer: D
44. A large department store examined a sample of the 18 credit card sales and recorded the amounts
charged for each of three types of credit cards: MasterCard, Visa and Discover. Six MasterCard sales,
seven Visa and five Discover sales were recorded. The store used ANOVA to test if the mean sales for
each credit card were equal. What are the degrees of freedom for the F statistic?
A) 18 in the numerator, 3 in the denominator
B) 3 in the numerator, 18 in the denominator
C) 2 in the numerator, 15 in the denominator
D) 6 in the numerator, 15 in the denominator
Answer: C
45. Suppose that an automobile manufacturer designed a radically new lightweight engine and wants to
recommend the grade of gasoline that will have the best fuel economy. The four grades are: regular,
below regular, premium, and super premium. The test car made three trial runs on the test track using
each of the four grades and the miles per gallon recorded. At the 0.05 level, what is the critical value of F
used to test the hypothesis that the miles per gallon for each fuel is the same.
Regular
39.31
39.87
39.87
A)
B)
C)
D)
Kilometers per liter
Below Regular Premium
36.69
38.99
40.00
40.02
41.01
39.99
Super Premium
40.04
39.89
39.93
1.96
4.07
2.33
12.00
Answer: B
46. Three different fertilizers were applied to a field of celery. In computing F, how many degrees of
freedom are there in the numerator?
A) 0
B) 1
C) 2
D) 3
Answer: C
47. Suppose a package delivery company purchased 14 trucks at the same time. Five trucks were
purchased from manufacturer A, four from B and five from manufacturer C. The cost of maintaining
each truck was recorded. The company used ANOVA to test if the mean maintenance cost of the trucks
from each manufacturer were equal. To apply the F test, how many degrees of freedom are in the
denominator?
A) 2
B) 3
C) 11
D) 14
Answer: C
48. In an effort to determine the most effective way to teach safety principles to a group of employees,
four different methods were tried. Some employees were given programmed instruction booklets and
worked through the course at their own pace. Other employees attended lectures. A third group watched
a television presentation, and a fourth group was divided into small discussion groups. A high of 10 was
possible. A sample of five tests was selected from each group. The test grade results were:
Sample Number
1
2
3
4
5
Programmed Instruction
6
7
6
5
6
Lecture
8
5
8
6
8
TV
7
9
6
8
5
Group Discussion
8
5
6
6
5
At the 0.01 level, what is the critical value?
A) 1.00
B) 1.96
C) 3.24
D) 5.29
Answer: D
49. In ANOVA, an F statistic is used to test a null hypothesis such as:
A) H o : σ 21  σ 2 2  σ 2 3
B) H o : σ 21  σ 2 2   2 3
C) H o : μ 1  μ 2  μ 3
D) H o : μ 1  μ 2  μ 3
Answer: C
50. An electronics company wants to compare the quality of their cell phones to the cell phones from
three competitors. They sample 10 phones from each company and count the number of defects for each
phone. If ANOVA were used to compare the average number of defects, the treatments would be defined
as:
A) the number of cell phones sampled.
B) the average number of defects.
C) The total number of phones
D) The four companies.
Answer: D
51. Several employees have submitted different methods of assembling a subassembly. Sample data for
each method are:
Sample Number
1
2
3
Lind's Method
16.6
17.0
16.9
Minutes Required for Assembly
Szabo's Method Carl's Method
22.4
31.4
21.5
33.4
22.6
30.1
Manley's Method
18.4
19.6
17.6
How many treatments are there?
A) 3
B) 4
C) 12
D) 0
Answer: B
52. If an ANOVA test is conducted and the null hypothesis is rejected, what does this indicate?
A) Too many degrees of freedom
B) No difference between the population means
C) A difference between at least one pair of population means
D) None of the above
Answer: C
Scrambling: Locked
53. A preliminary study of hourly wages paid to unskilled employees in three metropolitan areas was
conducted. Seven employees were included from Area A, 9 from Area B and 12 from Area C. The test
statistic was computed to be 4.91. What can we conclude at the 0.05 level?
A) Mean hourly wages of unskilled employees all areas are equal
B) Mean hourly wages in at least 2 metropolitan areas are different
C) More degrees of freedom are needed
D) None of these is correct
Answer: B
Scrambling: Locked
54. In ANOVA analysis, when the null hypothesis is rejected, we can find which means are different by
A) constructing confidence intervals.
B) adding another treatment.
C) doing an additional ANOVA.
D) doing a t test.
Answer: A
55. In a two-way ANOVA, a blocking variable is used to
A) increase the error sum of squares.
B) decrease the error sum of squares.
C) increase the treatment sum of squares.
D) decrease the treatment sum of squares.
Answer: B
56. In a two-way ANOVA with interaction, a significant interaction term indicates that
A) the response variable is interactive.
B) a blocking factor is present.
C) both factors are unrelated.
D) both factors have a combined effect on the response variable.
Answer: D
Fill-in-the-Blank
57. The F distribution is a ______________ distribution.
Answer: continuous
58. What is the shape of the F distribution? ______________________
Answer: positively skewed
59. What are the minimum and maximum of values of an F distribution? _______ and _______
Answer: zero and positive infinity
60. What kind of values can the F distribution NOT have? ______________
Answer: negative values
61. When comparing two population variances we use the ___________ distribution.
Answer: F
62. What test statistic is used in ANOVA? ________________
Answer: F statistic
63. The calculated F value must be equal to or greater than _________ .
Answer: zero (0)
64. What test statistic is used to compare two variances? ________________
Answer: F statistic
65. The F-distribution is useful when testing a requirement of two-sample tests of hypothesis. What is
the assumption? ________________
Answer: The population variances are equal
66. What is the statistical technique used to test the equality of three or more population means called?
______________________
Answer: analysis of variance (ANOVA)
67. ANOVA requires that the populations should be _______, _______, and ______.
Answer: normal or normally distributed; independent, equal standard deviations or variances
68. What statistical technique is used to test the equality of three or more population means?
____________________
Answer: analysis of variance (ANOVA)
69. What is the least number of sources of variation in ANOVA? _________
Answer: two
70. In an ANOVA without a block source of variation, what are the degrees of freedom associated with
the error sum of squares? ___________
Answer: n - k
71. ANOVA, how many degrees of freedom are associated with the numerator of the F ratio? _______
Answer: k - 1 or b - 1 or (k - 1 )( b - 1 )
72. What equals the sum of squares divided by its corresponding degrees of freedom?
_________________
Answer: mean square
73. In ANOVA, what is the numerator of the F ratio called? ______________
Answer: treatment mean square, block mean square, or interaction mean square
74. Assuming that the larger of two variances is in the numerator of an F statistic, in which tail of the F
distribution is the rejection region for analysis of variance? ________
Answer: upper
75. In ANOVA, when we do not reject the null hypothesis, what inference do we make about the
population means? ________________
Answer: they are equal
76. What is the null hypothesis for an ANOVA? ____________________
Answer: H o : μ 1  μ 2  μ 3
77. When H0 is rejected in ANOVA, _______ _______ are constructed to identify means that differ.
Answer: confidence intervals
78. When a second treatment is included in the ANOVA analysis without interaction, that treatment is
called a __________________.
Answer: blocking variable
79. How many sources of variation are summarized in a two-way ANOVA table? _________
Answer: three
80. In a two-way ANOVA table, what are the error degrees of freedom? _________
Answer: (k - 1)(b - 1)
81. In a two-way ANOVA with interaction, table, what are the error degrees of freedom? _________
Answer: (n – kb)
82. In a two-way ANOVA with interaction, what are the interaction degrees of freedom? _________
Answer: (k - 1)(b - 1)
83. In a one-way ANOVA, what are the two sources of variation? _________
Answer: treatment and error variation
Goal: 5
84. In a two-way ANOVA without interaction, what are the three sources of variation?
Answer: Treatment, Block and error variation
Goal: 7
85. In a two-way ANOVA with interaction, what are the four sources of variation? _________
Answer: Factor A, factor B, interaction of factors A and B, error
Multiple Choice
Use the following to answer questions 86-96:
A manufacturer of automobile transmissions uses three different processes. The management ordered a
study of the production costs to see if there is a difference among the three processes. A summary of the
findings is shown below.
Process Totals ($ 100’s)
Sample Size
Sum of Squares
Process 1
137
10
1893
Process 2
108
10
1188
86. What is the sum of squares for the treatment?
A) 67.80
B) 58.07
C) 149.34
D) 23.47
Answer: B
87. What is the sum of squares of the error?
A) 67.80
B) 58.07
C) 149.34
D) 23.47
Answer: A
88. What is the critical value of F at the 5% level of significance?
A) 19.45
B) 3.00
C) 3.35
D) 3.39
Answer: C
89. What is the critical value of F at the 1% level of significance?
A) 99.46
B) 5.49
C) 5.39
D) 4.61
Answer: B
Process 3
107
10
1175
Total
352
30
4256
90. What are the degrees of freedom for the treatment sum of squares?
A) 2
B) 3
C) 10
D) 27
Answer: A
91. What are the degrees of freedom for the error sum of squares?
A) 3
B) 10
C) 27
D) 30
Answer: C
92. What are the total degrees of freedom?
A) 27
B) 28
C) 29
D) 30
Answer: C
93. What is the mean square for treatments?
A) 2.511
B) 2.151
C) 33.9
D) 29.035
Answer: D
94. What is the mean square for error?
A) 2.511
B) 2.151
C) 33.9
D) 29.035
Answer: A
95. What is the calculated F?
A) 0.086
B) 1.168
C) 11.56
D) 13.50
Answer: C
96. What is the decision?
A) Reject H0 -- there is a difference in treatment means
B) Fail to reject H0 -- there is a difference in treatment means
C) Reject H0 -- there is a difference in errors
D) Fail to reject H0 -- there is a difference in errors
Answer: A
Fill-in-the-Blank
Use the following to answer questions 97-106:
In a study of low tar cigarettes, five cigarettes from each of three brands were tested to see if the mean
amount of tar per cigarette differs among the brands.
97. What are the degrees of freedom for the numerator? ______
Answer: 2
98. What are the degrees of freedom for the denominator? ______
Answer: 12
99. If the sum of squares for the brands is 0.07, what is the mean square for brands? ______
Answer: 0.035
100. If the sum of squares for the error is 0.09, what is the mean square for the error? ______
Answer: 0.0075
101. What is the F critical value for á = 0.05? ______
Answer: 3.89
102. What is the calculated value of F if the brand sum of squares is 0.07 and the error sum of squares is
0.09? ______
Answer: 4.66
103. If F calculated is 4.75 what is the decision if  = 0.05? ___________
Answer: reject H0
104. If the calculated F is 4.74, what would the decision be if  = 0.01? _________________
Answer: Do not reject H0
105. If the sum of squares for the brands is 0.05 and the sum of squares for the error is 0.09, what is the
decision rule if α = 0.05? ________________________
Answer: Do not reject H0 since calculated F = 3.33 and F(0.05) = 3.89
106. If the sum of squares for the brands is 0.07 and the sum of squares for the error is 0.11, what is the
decision rule at α = 0.05? __________________________
Answer: Do not reject H0 since calculated F = 3.88 and F(0.05) = 3.89
Multiple Choice
Use the following to answer questions 107-111:
Given the following Analysis of Variance table for three treatments each with six observations.
Source
Treatments
Error
Total
Sum of Squares
1116
1068
2184
df
Mean Square
107. What are the degrees of freedom for the numerator and denominator?
A) 3 and 18
B) 2 and 17
C) 3 and 15
D) 2 and 15
Answer: D
108. What is the critical value of F at the 5% level of significance?
A) 3.29
B) 3.68
C) 3.59
D) 3.20
Answer: C
109. What is the mean square for treatments?
A) 71.2
B) 71.4
C) 558
D) 534
Answer: C
110. What is the computed value of F?
A) 7.48
B) 7.84
C) 8.84
D) 8.48
Answer: B
111. What is the decision?
A) Reject H0 -- there is a difference in treatment means
B) Fail to reject H0 -- there is a difference in treatment means
C) Reject H0 -- there is a difference in errors
D) Fail to reject H0 -- there is a difference in errors
Answer: A
Fill-in-the-Blank
Use the following to answer questions 112-123:
A bottle cap manufacture with four machines and six operators wants to see if variation in production is
due to the machines and/or the operators. Each operator is assigned to each machine with the following
Analysis of Variance table.
Source
Machines
Operators
Error
Total
Sum of Squares
114
215
54
383
df
Mean Square
112. What are the degrees of freedom for the machines? ______
Answer: 3
113. What are the degrees of freedom for the operators? _____
Answer: 5
114. What are the degrees of freedom for the errors? _____
Answer: 15
115. What is the critical value of F for the machine treatment effect at the 1% level of significance? ____
Answer: 5.42
116. What is the critical value of F for the operator block effect at the 1% level of significance? ____
Answer: 4.56
117. What is the mean square for machines? _____
Answer: 111.39
118. What is the mean square for operators? _____
Answer: 47.44
119. What is the mean square for errors? _____
Answer: 24.77
120. What is the computed value of F for the machines? _____
Answer: 4.50
121. What is the computed value of F for the operators? _____
Answer: 1.92
Essay
122. Using a 1% significance level, what is the decision for the machines?
__________________________
Answer: Do not reject H0; there is no difference in production based on machines
123. Using a 1% level of significance, what is the decision for the operators?
__________________________
Answer: Do not reject H0; there is no difference in production based on the operators
Multiple Choice
Use the following to answer questions 124-132:
Two accounting professors decided to compare the variation of their grading procedures. To accomplish
this they each graded the same 10 exams with the following results:
Professor 1
Professor 2
124. What is H0?
A) σ 21  σ 2 2
Mean Grade
79.3
82.1
Standard Deviation
22.4
12.0
B) σ 21  σ 2 2
C) μ1  μ 2
D) μ1  μ 2
Answer: A
125. What is H1?
A) σ 21  σ 2 2
B) σ 21  σ 2 2
C) μ1  μ 2
D) μ1  μ 2
Answer: B
126. What are the degrees of freedom for the numerator of the F ratio?
A) 8
B) 9
C) 10
D) 18
E) 20
Answer: B
127. What are the degrees of freedom for the denominator of the F ratio?
A) 20
B) 18
C) 10
D) 9
E) 8
Answer: D
128. What is the critical value of F at the 0.01 level of significance?
A) 5.85
B) 5.35
C) 6.51
D) 4.03
Answer: B
129. What is the critical value of F at the 0.05 level of significance?
A) 5.85
B) 5.35
C) 3.18
D) 4.03
Answer: C
130. The calculated F ratio is
A) 3.484
B) 1.867
C) 3.18
D) 5.35
Answer: A
131. At the 1% level of significance, what is the decision?
A) Reject the null hypothesis and conclude the variance is different.
B) Fail to reject the null hypothesis and conclude the variance is different.
C) Reject the null hypothesis and conclude the variance is the same.
D) Fail to reject the null hypothesis and conclude the variance is the same.
Answer: D
132. At the 5% level of significance, what is the decision?
A) Reject the null hypothesis and conclude the variance is different.
B) Fail to reject the null hypothesis and conclude no significant difference in the variance.
C) Reject the null hypothesis and conclude the variance is the same.
D) Fail to reject the null hypothesis and conclude the variance is the same.
Answer: A
Use the following to answer questions 133-136:
A random sample of 30 executives from companies with assets over $1 million was selected and asked
for their annual income and level of education. The ANOVA comparing the average income among three
levels of education rejected the null hypothesis. The Mean Square Error (MSE) was 243.7. The
following table summarized the results:
Number sampled
Mean salary (1,000’s)
High School
or Less
7
49
Undergraduate
Degree
11
76.3
Master’s Degree
or More
12
78.3
133. When comparing the mean salaries to test for differences between treatment means, the t statistic is
based on:
A) The treatment degrees of freedom.
B) The total degrees of freedom.
C) The error degrees of freedom
D) The ratio of treatment and error degrees of freedom
Answer: C
134. When comparing the mean annual incomes for executives with Undergraduate and Master's Degree
or more, the following 95% confidence interval can be constructed:
A) 2.0  2.052*6.51
B) 2.0  3.182*6.51
C) 2.0  2.052*42.46
D) None of the above
Answer: A
135. Based on the comparison between the mean annual incomes for executives with Undergraduate and
Master's Degree or more,
A) A confidence interval shows that the mean annual incomes are not significantly different.
B) The ANOVA results show that the mean annual incomes are significantly different.
C) A confidence interval shows that the mean annual incomes are significantly different.
D) The ANOVA results show that the mean annual incomes are not significantly different.
Answer: A
136. When comparing the mean annual incomes for executives with a High School education or less and
Undergraduate Degree, the 95% confidence interval shows an interval of 11.7 to 42.7 for the difference.
This result indicates that
A) There is no significant difference between the two incomes.
B) The interval contains a difference of zero.
C) Executives with and Undergraduate Degree earn significantly more than executives with a High
School education or less.
D) Executives with and Undergraduate Degree earn significantly less than executives with a High School
education or less.
Answer: C
Fill-in-the-Blank
Use the following to answer questions 137-153:
A bottle cap manufacture with four machines and three operators wants to see if variation in hourly
production is due to the machines and/or the operators or an interaction effect of machine and operator.
Each operator is assigned to each machine and the production of caps from 3 randomly selected hours is
recorded. The analysis shows the following Analysis of Variance table.
137. What are the degrees of freedom for the machines? ______
Answer: 3
138. What are the degrees of freedom for the operators? _____
Answer: 2
139. What are the degrees of freedom for the interaction? _____
Answer: 6
140. What are the degrees of freedom for the error? _____
Answer: 24
141. What is the critical value of F for the machine effect at the 1% level of significance? ____
Answer: 4.72
142. What is the critical value of F for the operator effect at the 1% level of significance? ____
Answer: 5.61
143. What is the critical value of F for the interaction effect at the 1% level of significance? ____
Answer: 3.67
144. What is the mean square for machines? _____
Answer: 38
145. What is the mean square for operators? _____
Answer: 57.5
146. What is the mean square for interaction? _____
Answer: 16.67
147. What is the mean square for errors? _____
Answer: 2.25
148. What is the computed value of F for the machines? _____
Answer: 16.89
149. What is the computed value of F for the operators? _____
Answer: 25.56
150. What is the computed value of F for the interaction? _____
Answer: 7.41
151. Using a 1% significance level, what is the decision for the machines?
__________________________
Answer: Reject H0; there is a difference in production based on machines
Essay
152. Using a 1% level of significance, what is the decision for the operators?
__________________________
Answer: Reject H0; there is a difference in production based on the operators
153. Using a 1% level of significance, what is the decision for the interaction?
__________________________
Answer: Reject H0; there is an interaction effect of machine and operator