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Chapter 18—Analysis of variance
MULTIPLE CHOICE
1. In one-way ANOVA, the amount of total variation that is unexplained is measured by the:
A.
sum of squares for treatments.
B.
sum of squares for error.
C.
total sum of squares.
D.
number of degrees of freedom.
2. The test statistic of the single-factor ANOVA equals:
A.
sum of squares for treatments/sum of squares for error.
B.
sum of squares for error/sum of squares for treatments.
C.
mean square for treatments/mean square for error.
D.
mean square for error/mean square for treatments.
3. Which of the following statements is false?
A.
The sum of squares for treatments (SST) explains some of the variation.
B.
The sum of squares for error (SSE) measures the amount of variation that is unexplained.
C.
SS(Total) = SST + SSE
D.
SS(Total) measures the amount of variation within the samples.
4. In one-way ANOVA, suppose that there are four treatments with n1 = 7, n2 = 6, n3 = 5, and n4 = 7. Then
the rejection region for this test at the 5% level of significance is:
A.
F > F0.05,3,24.
B.
F > F0.025,3,21.
C.
F > F0.05,3,21.
D.
F > F0.025,3,24.
5. In an ANOVA test, the test statistic is F = 3.55. The rejection region is F > 3.07 for the 5% level of
significance, F > 3.82 for the 2.5% level, and F > 4.87 for the 1% level. For this test, the p-value is:
A.
greater than 0.05.
B.
between 0.025 and 0.05.
C.
between 0.01 and 0.025.
D.
smaller than 0.01.
6. In a two-tailed pooled-variance t-test (equal-variances t-test), the null and alternative hypotheses are
exactly the same as in one-way ANOVA with:
A.
exactly one treatment.
B.
exactly two treatments.
C.
exactly three treatments.
D.
any number of treatments.
7. Which of the following is not a required condition for one-way ANOVA?
A.
The sample sizes must be equal.
B.
The populations must all be normally distributed.
C.
The population variances must be equal.
D.
The samples for each treatment must be selected randomly and independently.
8. The following equation applies to which ANOVA model?
SS(Total) = SST + SSE.
A.
One-way ANOVA.
B.
Two-way ANOVA.
C.
Completely randomised design.
D.
Randomised block design.
9. The primary interest of designing a randomised block experiment is to:
A.
reduce the variation among blocks.
B.
increase the between-treatments variation to more easily detect differences among the
treatment means.
C.
reduce the within-treatments variation to more easily detect differences among the
treatment means.
D.
increase the total sum of squares.
10. Two independent samples of 20 each have been selected at random from the male and female students
of a large university. To test whether there is any difference in the grade point average between male
and female students, an equal-variances t-test will be considered. Another test to consider is ANOVA.
The most likely ANOVA to fit this test situation is the:
A.
one-way ANOVA.
B.
two-way ANOVA.
C.
randomised block design.
D.
chi-squared test.
11. The analysis of variance is a procedure that allows statisticians to compare two or more population:
A.
modes.
B.
proportions.
C.
means.
D.
standard deviations.
12. The distribution of the test statistic for analysis of variance is the:
A.
normal distribution.
B.
Student t distribution.
C.
F distribution.
D.
chi-squared distribution.
13. The simplest experimental design has:
A.
a single response variable.
B.
two response variables.
C.
three response variables.
D.
no response variables at all.
14. Which of the following is not true of the F-distribution?
A.
The mean and median are equal.
B.
It is skewed to the right.
C.
Its values are always positive.
D.
It is used in the ANOVA test.
15. In a single-factor analysis of variance, MST is the mean square for treatments and MSE is the mean
square for error. The null hypothesis of equal population means is likely false if:
A.
MST is much larger than MSE.
B.
MST is much smaller than MSE.
C.
MST is equal to MSE.
D.
MST is zero.
16. If we want to conduct a test to determine whether a population mean is greater than another population
mean, we:
A.
can use the analysis of variance.
B.
must use the independent-samples t-test for the difference between two means.
C.
must use the chi-squared test.
D.
Both A and B are correct.
17. In ANOVA, error variability is computed as the sum of the squared errors, SSE, for all values of the
response variable. This variability is the:
A.
total variation.
B.
within-group variation.
C.
between-groups variation.
D.
None of the above answers is correct.
18. In a one-way ANOVA where there are k treatments and n observations, the numbers of degrees of
freedom for the F-statistic are equal to:
A.
k – 1 and n – 1.
B.
k – 1 and n – k.
C.
n – k and k – 1.
D.
nk – 1 and k – 1.
19. One-way ANOVA is applied to three independent samples having means 10, 13 and 18, respectively.
If each observation in the first sample were decreased by 5, the value of the F-statistic would:
A.
increase.
B.
decrease.
C.
remain unchanged.
D.
decrease by 5.
20. The F-statistic in a one-way ANOVA represents the variation:
A.
between the treatments plus the variation within the treatments.
B.
within the treatments minus the variation between the treatments.
C.
between the treatments divided by the variation within the treatments.
D.
variation within the treatments divided by the variation between the treatments.
21. In the one-way ANOVA where k is the number of treatments and n is the number of observations in all
samples, the number of degrees of freedom for error is given by:
A.
k – 1.
B.
n – k.
C.
n – 1.
D.
n – k + 1.
22. In the one-way ANOVA where k is the number of treatments and n is the number of observations in all
samples, the number of degrees of freedom for treatments is given by:
A.
k – 1.
B.
n – k.
C.
n – 1.
D.
n – k + 1.
23. Three tennis players, one a beginner, one intermediate and one advanced, have been randomly selected
from the membership of a club in a large city. Using the same tennis ball, each player hits ten serves,
one with each of three racquet models, with the three racquet models selected randomly. The speed of
each serve is measured with a machine and the result recorded. Among the ANOVA models listed
below, the most likely model to fit this situation is the:
A.
one-way ANOVA.
B.
two-way ANOVA.
C.
randomised block design.
D.
matched-pairs model.
24. A survey will be conducted to compare the grade point averages of US high-school students from four
different school districts. Students are to be randomly selected from each of the four districts and their
grade point averages recorded. The ANOVA model most likely to fit this situation is:
A.
one-way ANOVA.
B.
two-way ANOVA.
C.
randomised block design.
D.
complete 4 4 factorial design.
25. In ANOVA, the F-test is the ratio of two sample variances. In the one-way ANOVA (completely
randomised design), the variance used as the denominator of the ratio is the:
A.
mean square for treatments.
B.
mean square for error.
C.
mean square for blocks.
D.
total sum of squares.
26. In the randomised block design ANOVA, the sum of squares for error equals:
A.
SS(Total) – SST.
B.
SS(Total) – SSB.
C.
SS(Total) – SST – SSB.
D.
SS(Total) – SS(A) – SS(B) – SS(AB).
27. When the effect of a level for one factor depends on which level of another factor is present, the most
appropriate ANOVA design to use in this situation is the:
A.
one-way ANOVA.
B.
two-way ANOVA.
C.
randomised block design.
D.
matched pairs design.
28. The randomised block design with exactly two treatments is equivalent to a two-tailed:
A.
independent samples z-test.
B.
independent-samples equal-variances t-test.
C.
independent-samples unequal-variances t-test.
D.
matched pairs t-test.
29. A randomised block design with 4 treatments and 5 blocks produced the following sum of squares
values: SS(Total) = 1951, SSB = 1414.4, SSE = 188. The value of SST must be:
A.
99.
B.
3553.
C.
349.
D.
None of the above is correct.
30. One-way ANOVA is performed on independent samples taken from three normally distributed
populations with equal variances. The following summary statistics were calculated:
=
1
n
7,
=
1
x
65,
=
1
s
4.2.
=
2
n
8,
=
2
x
59,
=
2
s
4.9.
=
3
n
9,
=
3
x
63,
=
3
s
4.6.
The value of the test statistics, F, equals:
A.
71.250.
B.
0.322.
C.
21.104.
D.
3.376.
31. In a completely randomised design for ANOVA, the numbers of degrees of freedom for the numerator
and denominator are 4 and 25, respectively. The total number of observations must equal:
A.
29.
B.
25.
C.
30.
D.
24.
32. The number of degrees of freedom for the denominator of a one-way ANOVA test for 4 population
means with 15 observations sampled from each population is:
A.
60.
B.
19.
C.
56.
D.
45.
33. In one-way ANOVA, the term
x
refers to the:
A.
sum of the sample means.
B.
sum of the sample means divided by the total number of observations.
C.
sum of the population means.
D.
weighted mean of the sample means.
34. One-way ANOVA is performed on three independent samples with n1 = 10, n2 = 8 and n3 = 9. The
critical value obtained from the F-table for this test at the 5% level of significance equals:
A.
39.46.
B.
3.40.
C.
4.32.
D.
19.45.
35. Which of the following is a correct formulation for the null hypothesis in one-way ANOVA?
A.
0
321 =++
.
B.
0
321 ++
.
C.
321
==
.
D.
321
.
36. One-way ANOVA is performed on independent samples taken from three normally distributed
populations with equal variances. The following summary statistics are calculated:
=
1
n
6,
=
1
x
50,
=
1
s
5.2.
=
2
n
8,
=
2
x
55,
=
2
s
4.9 .
=
3
n
6,
=
3
x
51,
=
3
s
5.4.
The grand mean equals:
A.
50.0.
B.
52.0.
C.
52.3.
D.
53.0.
37. One-way ANOVA is applied to independent samples taken from three normally distributed
populations with equal variances. The following summary statistics are calculated:
=
1
n
18,
=
1
x
15,
=
1
s
2.
=
2
n
10,
=
2
x
20,
=
2
s
3.
=
3
n
12,
=
3
x
16,
=
3
s
1.
The within-treatments variation equals:
A.
82.95.
B.
160.
C.
19.2.
D.
165.9.
38. Which of the following is not true of Tukey’s multiple comparison method?
A.
It is based on the studentised range statistic q to obtain the critical value needed to
construct individual confidence intervals.
B.
It requires that all sample sizes are equal, or at least similar.
C.
It can be employed instead of the analysis of variance.
D.
All of the above statements are true.
39. A professor of statistics at Wayne State University in the US wants to determine whether the average
starting salaries among graduates of the 15 universities in Michigan are equal. A sample of 25 recent
graduates from each university is randomly taken. The appropriate critical value for the ANOVA test
is obtained from the F-distribution with number so of degrees of freedom equal to:
A.
15 and 25.
B.
14 and 360.
C.
360 and 14.
D.
25 and 15.
40. One-way ANOVA is applied to independent samples taken from three normally distributed
populations with equal variances. The following summary statistics are calculated:
=
1
n
10,
=
1
x
40,
=
1
s
5.
=
2
n
10,
=
2
x
48,
=
2
s
6.
=
3
n
10,
=
3
x
50,
=
3
s
4.
The between-treatments variation equals:
A.
460.
B.
688.
C.
560.
D.
183.
41. One-way ANOVA is applied to independent samples taken from four normally distributed populations
with equal variances. If the null hypothesis is rejected, then we can infer that:
A.
all population means are equal.
B.
all population means differ.
C.
at least two population means are equal.
D.
at least two population means differ.
42. Consider the following partial ANOVA table:
Source of Variation
SS
df
MS
F
Treatments
75
*
25
6.67
Error
60
*
3.75
Total
135
19
The numbers of degrees of freedom for numerator and denominator, respectively, (identified by
asterisks) are:
A.
4 and 15.
B.
3 and 16.
C.
15 and 4.
D.
16 and 3.
43. In single-factor analysis of variance, between-treatments variation stands for the:
A.
sum of squares for error.
B.
sum of squares for treatments.
C.
total sum of squares.
D.
Both A and B are correct.
44. Consider the following ANOVA table:
Source of Variation
SS
df
MS
F
Treatments
4
2
2.0
0.80
Error
30
12
2.5
Total
34
14
The number of treatments is:
A.
13.
B.
5.
C.
3.
D.
33.
45. In one-way analysis of variance, within-treatments variation stands for the:
A.
sum of squares for error.
B.
sum of squares for treatments.
C.
total sum of squares.
D.
None of the above answers is correct.
46. Consider the following ANOVA table:
Source of Variation
SS
df
MS
F
Treatments
128
4
32
2.963
Error
270
25
10.8
Total
398
29
The number of observations in all samples is:
A.
25.
B.
29.
C.
30.
D.
32.
47. In one-way analysis of variance, if all the sample means are equal, then the:
A.
total sum of squares is zero.
B.
sum of squares for error is zero.
C.
sum of squares for treatments is zero.
D.
sum of squares for error equals sum of squares for treatments.
48. In single-factor analysis of variance, if large differences exist among the sample means, it is then
reasonable to:
A.
reject the null hypothesis.
B.
reject the alternative hypothesis.
C.
fail to reject the null hypothesis.
D.
None of the above answers is correct.
49. Which of the following is not a required condition for one-way ANOVA?
A.
The populations are normally distributed.
B.
The population variances are equal.
C.
The samples are selected independently of each other.
D.
The population means are equal.
50. In one-way ANOVA, suppose that there are five treatments with
5
321 === nnn
and
7
54 == nn
.
Then the mean square for error, MSE, equals:
A.
SSE / 4.
B.
SSE / 29.
C.
SSE / 24.
D.
SSE / 5.
TRUE/FALSE
1. Statistics practitioners use the analysis of variance (ANOVA) technique to compare two or more
populations of interval data.
2. Given the significance level 0.025, the F-value for the numbers of degrees of freedom d.f. = (12, 21) is
2.64.
3. Three tennis players, one a beginner, one experienced and one a professional, have been randomly
selected from the membership of a large city tennis club. Using the same ball, each person hits four
serves with each of five racquet models, with the five racquet models selected randomly. Each serve is
clocked with a radar gun and the result recorded. Among ANOVA models, this setup is most like the
simple regression model.
4. Given the significance level 0.05, the F-value for the numbers of degrees of freedom d.f. = (9, 6) is
4.10.
5. Three tennis players, one a beginner, one experienced and one a professional, have been randomly
selected from the membership of a large city tennis club. Using the same ball, each person hits four
serves with each of five racquet models, with the five racquet models selected randomly. Each serve is
clocked with a radar gun and the result recorded. Among ANOVA models, this setup is most like the
randomised block design.
6. The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample means
are equal.
7. The analysis of variance (ANOVA) technique analyses the variance of the data to determine whether
differences exist between the population means.
8. In ANOVA, the between-treatments variation is denoted by SST, which stands for the total sum of
squares.
9. A study is to be undertaken to examine the effects of two kinds of background music and of two
assembly methods on the output of workers at a fitness shoe factory. Two workers will be randomly
assigned to each of four groups, for a total of eight in the study. Each worker will be given a
headphone set so that the music type can be controlled. The number of shoes completed by each
worker will be recorded. Does the kind of music or the assembly method or a combination of music
and method affect output? The ANOVA model most likely to fit this situation is the two-way analysis
of variance.
10. The sum of squares for error is also known as the between-treatments variation.
11. Two samples of 10 each have been taken from the male and female workers of a large company. The
data involve the wage rate of each worker. To test whether there is any difference in the average wage
rate between male and female workers, a pooled-variances t-test will be considered. Another test
option to consider is ANOVA. The most likely ANOVA to fit this test situation is the randomised
block design.
12. We do not need the t-test of
1 2
−
, since the analysis of variance can be used to test the difference
between the two population means.
13. Conceptually and mathematically, the F-test of the independent-samples single-factor ANOVA is an
extension of the t-test of
1 2
−
.
14. When the problem objective is to compare more than two populations, the experimental design that is
the counterpart of the matched pairs experiment is called one-way analysis of variance.
15. In employing the randomised block design, the primary interest lies in reducing the within-treatments
variation in order to make easier to detect differences between the treatment means.
16. In ANOVA, a factor is an independent variable.
17. When the data are obtained through a controlled experiment in the single-factor ANOVA, we call the
experimental design the completely randomised design of the analysis of variance.
18. In one-way ANOVA, the total variation SS(Total) is partitioned into three sources of variation: the
sum of squares for treatments (SST), the sum of squares for blocks (SSB) and the sum of squares for
error (SSE).
19. If we examine two or more independent samples to determine if their population means could be
equal, we are performing one-way analysis of variance (ANOVA).
20. The randomised block design is also called the two-way analysis of variance.
21. The purpose of designing a randomised block experiment is to reduce the between-treatments variation
(SST) to more easily detect differences between the treatment means.
22. The F-test of the analysis of variance requires that the populations be normally distributed with equal
variances.
23. The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample sizes
are equal.
24. The F-test of the randomised block design of the analysis of variance has the same requirements as the
independent-samples design; that is, the random variable must be normally distributed and the
population variances must be equal.
25. A randomised block experiment having 5 treatments and 6 blocks produced the following values:
SST = 252, SSB = 1095, SSE = 198. The value of SS(Total) must be 645.
26. In one-way ANOVA, the test statistic is defined as the ratio of the mean square for error (MSE) and
the mean square for treatments (MST), that is, F = MSE / MST.
27. When the response is not normally distributed, we can replace the randomised block ANOVA with its
non-parametric counterpart; the Friedman test.
28. If we first arrange test units into similar groups before assigning treatments to them, the test design we
should use is the randomised block design.
29. If the data are not normally distributed, we can replace the independent-samples single-factor model of
the analysis of variance with its non-parametric counterpart, which is the Kruskal–Wallis test.
30. The sum of squares for treatments (SST) is the variation attributed to the differences between the
treatment means, while the sum of squares for error (SSE) measures the variation within the samples.
31. The calculated value of F in a one-way analysis is 7.88. The numbers of degrees of freedom for
numerator and denominator are 3 and 9, respectively. The most accurate statement to be made about
the p-value is that p-value < 0.01.
32. The number of degrees of freedom for the numerator or MST is 3 and that for the denominator or MSE
is 18. The total number of observations in the completely randomised design must equal 20.
33. A survey is to be conducted to compare the superannuation contributions made by employees from
three Victorian universities. Employees are to be randomly selected from each of the three universities
and the dollar amounts of their contributions recorded. The ANOVA model most likely to fit this
situation is the randomised block design.
34. Given the significance level 0.025, the F-value for the numbers of degrees of freedom d.f. = (4, 8) is
8.98.
35. One-way ANOVA is applied to three independent samples having means 12, 15 and 20, respectively.
If each observation in the third sample were increased by 40, the value of the F-statistic would
increase by 40.
36. The F-statistic in a one-way ANOVA represents the variation within the treatments divided by the
variation between the treatments.
37. The sum of squares for error (SSE) explains some of the total variation, while the sum of squares for
treatments (SST) measures the amount of variation that is unexplained.
38. The distribution of the test statistic for analysis of variance is the F-distribution.
39. In one-way ANOVA, suppose that there are 5 treatments with
1 2 3
n n n= = =
5 and
4 5
n n= =
8. Then
the mean square for error, MSE, equals SSE/26.
40. A randomised block design with 4 treatments and 5 blocks produced the following sum of squares
values: SS(Total) = 2000, SST = 400, SSE = 200. The value of MSB must be 350.
41. The number of degrees of freedom for the denominator of a one-way ANOVA test for 5 population
means with 12 observations sampled from each population is 60.
42. Tukey’s multiple comparison method determines a critical number,
; such that if any pair of sample
means has a difference smaller than
, we conclude that the pair’s two corresponding population
means are different.
SHORT ANSWER
1. The following data are drawn from three normal populations.
Treatment
1
2
3
8
11
16
16
13
20
13
12
15
14
15
13
9
10
11
Set up the ANOVA table and test at the 5% level of significance to determine whether differences
exist among the population means.
2. Provide an example for a randomised block design with three treatments (k = 3) and four blocks (b =
4), in which SST is equal to zero and SSB and SSE are not equal to zero.
3. A randomised block design experiment produced the following data.
Treatment
Block
1
2
3
1
8
17
5
2
5
10
12
3
9
14
13
4
12
11
13
5
12
14
16
a. Test to determine whether the treatment means differ. (Use
= 0.05)
b. Test to determine whether the block means differ. (Use
= 0.05)
