ANOVA PROBLEMS
1. A company conducted a consumer research project to ascertain customer
service ratings from its customers. The customers were asked to rate the
company on a scale from 1 to 7 on various quality characteristics. One question
was the promptness of company response to a repair problem. The following
data represent customer responses to this question. The customers were divided
by geographic region and by age. Use analysis of variance to analyze the
responses. Let α = 0.05. Graph the cell means and observe any interaction.
Geographic Region
Age
Southeast
West
Midwest
Northeast
3
2
3
2
21 – 35
2
4
3
3
3
3
2
2
5
4
5
6
36 – 50
5
4
6
4
4
6
5
5
3
2
3
3
Over 50
1
2
2
2
2
3
3
1
2. A major automobile manufacturer wants to know whether there is any difference
in the average mileage of four different brands of tires (A, B, C, and D), because
the manufacturer is trying to select the best supplier in terms of tire durability.
The manufacturer selects comparable levels of tires from each company and
tests some on comparable cars. The mileage results follow.
A
B
C
D
31,000
24,000
30,500
24,500
25,000
25,500
28,000
27,000
28,500
27,000
32,500
26,000
29,000
26,500
28,000
21,000
32,000
25,000
31,000
25,500
27,500
28,000
26,000
27,500
Use α = 0.05 to test whether there is a significant difference in the mean mileage
of these four brands. Assume tire mileage is normally distributed.
3. Agricultural researchers are studying three different ways of planting peanuts to
determine whether significantly different levels of production yield will result. The
researchers have access to a large peanut farm on which to conduct their tests.
They identify six blocks of land. In each block of land, peanuts are planted in
each of the three different ways. At the end of the growing season, the peanuts
are harvested and the average number of pounds per acre is determined for
peanuts planted under each method in each block. Using the following data and
α = 0.01, test to determine whether there is a significant difference in yields
among the planting methods.
Block
Method 1
Method 2
Method 3
1
1310
1080
850
2
1275
1100
1020
3
1280
1050
780
4
1225
1020
870
5
1190
990
805
6
1300
1030
910
4. The Construction Labor Research Council lists a number of construction labor
jobs that seem to pay approximately the same wages per hour. Some of these
are bricklaying, iron working, and crane operation. Suppose a labor researcher
takes a random sample of workers from each of these types of construction jobs
and from across the country and asks what their hourly wages are. If this survey
yields the following data, is there a significant difference in mean hourly wages
for these three jobs? Let α = 0.05
Job Type
Bricklaying
Iron Working
Crane Operation
19.25
26.45
16.20
17.80
21.10
23.30
20.50
16.40
22.90
24.33
22.86
19.50
19.81
25.55
27.00
22.29
18.50
22.95
21.20
25.52
21.20
5. Why are mergers attractive to CEOs? One of the reasons might be a potential
increase in market share that can come with the pooling of company markets.
Suppose a random survey of CEOs is taken, and they are asked to respond on a
scale from 1 to 5 (5 representing strongly agree) whether increase in market
share is a good reason for considering a merger of their company with another.
Suppose also that the data are as given here and that CEOs have been
categorized by size of company and years they have been with their company.
Use a two-way ANOVA to determine whether there are any significant
differences in the responses to this question. Use α = 0.05.
Company Size ($ million per year in sales)
Years with the company
0–5
6 – 20
21 – 100
> 100
2
2
3
3
0–2
3
1
4
4
2
2
4
4
2
3
5
3
2
2
3
3
3–5
1
3
2
3
2
2
4
3
3
3
4
4
2
2
3
2
Over 5
1
3
2
3
1
1
3
2
2
2
3
3
6. Are some office jobs viewed as having more status than others? Suppose a
study is conducted in which eight unemployed people are interviewed. The
people are asked to rate each of five positions on a scale from 1 to 10 to indicate
the status of the position, with 10 denoting most status and 1 denoting least
status. The resulting data are given below. Use α = 0.05 to analyze the repeated
measures randomized block design data.
Job
Respondent
Data
RecepAdmin.
Mail Clerk
Entry
tionist
Secretary
Asst.
1
4
5
3
7
6
2
2
4
4
5
4
3
3
3
2
6
7
4
4
4
4
5
4
5
3
5
1
3
5
6
3
4
2
7
7
7
2
2
2
4
4
8
3
4
3
6
6
7. An article in the Journal of Testing and Evaluation (1988, Vol. 16, pp. 508-515)
investigated the effects of cyclic loading frequency and environment conditions
on fatigue crack growth at a constant 22 MPa stress for a particular material. The
data from the experiment follow. The response is fatigue crack growth rate.
Environment
Air
H2O
Salt H2O
2.29
2.06
1.90
2.47
2.05
1.93
10
2.48
2.23
1.75
2.12
2.03
2.06
2.65
3.20
3.10
2.68
3.18
3.24
Frequency 1
2.06
3.96
3.98
2.38
3.64
3.24
2.24
11.00
9.96
2.71
11.00
10.01
0.1
2.81
9.06
9.36
2.08
11.30
10.40
Is there indication that either factor affects crack growth rate? Is there any
indication of interaction? Use α = 0.05.
8. An article in Lubrication Engineering (December, 1990) described the results of
an experiment designed to investigate the effects of carbon material properties
on the progression of blisters on carbon face seals. The carbon face seals are
used extensively in equipment such as air turbine starters. Five different carbon
materials were tested, and the surface roughness was measured. The data are
as follows:
Carbon
Material
Type
Surface Roughness
EC 10
0.50
0.55
0.55
0.36
EC 10A
0.31
0.07
0.25
0.18
0.56
0.20
EC4
0.20
0.28
0.12
EC1
0.10
0.16
Does carbon material type have an effect on mean surface roughness? Use α =
0.05. Find the P-value of the test.
9. In “The Effect of Nozzle Design on the Stability and Performance of Turbulent
Water Jets” (Fire Safety Journal, Vol. 4, August 1981), C. Theobald described an
experiment in which a shape measurement was determined for several different
nozzle types at different levels of jet efflux velocity. Interest in this experiment
focuses primarily on nozzle types, and velocity is a nuisance factor. The data are
as follows:
Nozzle
Jet Efflux Velocity (m/s)
Type
11.73
14.37
16.59
20.43
23.46
28.74
1
0.78
0.80
0.81
0.75
0.77
0.78
2
0.85
0.85
0.92
0.86
0.81
0.83
3
0.93
0.92
0.95
0.89
0.89
0.83
4
1.14
0.97
0.98
0.88
0.86
0.83
5
0.97
0.86
0.78
0.76
0.76
0.75
Does nozzle type affect the shape measurement? Use α = 0.05
10. An article in Nature describes an experiment to investigate the effect on
consuming chocolate on cardiovascular health (“Plasma Antioxidants from
Chocolate,” Vol. 424, 2003, pp. 1013). The experiment consisted of using three
different types of chocolate: 100 g of dark chocolate (DC), 100 g of dark
chocolate with 200 ml of full-fat milk (DC + MK), and 200 g of milk chocolate
(MC). Twelve subjects were used, seven women and five men, with an average
age range of 32.2 ± 1 years, and average weight of 65.8 ± 3.1 kg, and bodymass index of 21.9 ± 0.4 kg m-2. On different days, a subject consumed one of
the chocolate-factor levels, and one hour later the total antioxidant capacity of
their blood plasma was measured in an assay. Data similar to those summarized
in the article are shown below.
Subjects (Observations)
1
2
3
4
5
6
DC
118.8
122.6
115.6
113.6
119.5
115.9
DC + MK
105.4
101.1
102.7
97.1
101.9
98.9
MC
102.1
105.8
99.6
102.7
98.8
100.9
Subjects (Observations)
7
8
9
10
11
12
DC
115.8
115.1
116.9
115.4
115.6
107.9
DC + MK
100.0
99.8
102.6
100.9
104.5
93.5
MC
102.8
98.7
94.7
97.8
99.7
98.6
Analyze the experimental data using ANOVA. If α = 0.01, what conclusions
would you draw? Find the P-value of the test. Estimate the variability due to
random error.
11. An article in Quality Engineering [“Estimating Sources of Variation: A Case Study
from Polyurethane Product Research” (1999-2000, Vol. 12, pp. 89-96)] studied
the effects of additives on final polymer properties. In this case, polyurethane
additives were referred to as cross-linkers. The average domain spacing was the
measurement of the polymer property. The data are as follows:
Cross-Linker
Level
Domain Spacing (nm)
‒1
8.2
8.0
8.2
7.9
8.1
8.0
‒0.75
8.3
8.4
8.3
8.2
8.3
8.1
‒0.5
8.9
8.7
8.9
8.4
8.3
8.5
0
8.5
8.7
8.7
8.7
8.8
8.8
0.5
8.8
9.1
9.0
8.7
8.9
8.5
1
8.6
8.5
8.6
8.7
8.8
8.8
Is there a difference in the cross-linker level? Perform an ANOVA using α = 0.05.
Find the P-value of the test. Estimate the variability due to random error.
12. An experiment was conducted to investigate leaking current in a SOS MOSFETS
device. The purpose of the experiment was to investigate how leakage current
varies as the channel length changes. Four channel lengths were selected. For
each channel length, five different widths were also used, and width is to be
considered a nuisance factor. The data are as follows:
Channel
Width
Length
1
2
3
4
5
1
0.7
0.8
0.7
0.9
1.0
2
0.8
0.8
0.9
0.9
1.0
3
0.9
1.0
1.7
1.9
4.0
4
1.0
1.6
2.0
3.0
20.0
a) Test the hypothesis that mean leakage voltage does not depend on the
channel length, using α = 0.05.
b) The observed leakage voltage for channel length 4 and width 5 was
erroneously recorded. The correct observation is 4.0. Analyze the
corrected data from this experiment. Is there evidence to conclude that
mean leakage voltage increases with channel length?
13. In the book Design and Analysis of Experiments, 7th edition (2009, John Wiley &
Sons), the results of an experiment involving a storage battery used in the
launching mechanism of a shoulder-fired ground-to-air missile were presented.
Three material types can be used to make the battery plates. The objective is to
design a battery that is relatively unaffected by the ambient temperature. The
output response from the battery is effective life in hours. Three temperature
levels are selected, and a factorial experiment with four replicates is run. The
data are as follows:
Temperature (oF)
Material
Low
Medium
High
1
130
155
34
40
20
70
74
180
80
75
82
58
2
150
188
136
122
25
70
159
126
106
115
58
45
3
138
110
174
120
96
104
168
160
150
139
82
60
Test the appropriate hypotheses and draw conclusions using ANOVA with α =
0.05.
14. An experiment was run to determine whether four specific firing temperatures
affect the density of a certain type of brick. The experiment led to the following
data.
Temp.
(oC)
Density
38
21.8
21.9
21.5
21.6
21.7
21.5
21.8
52
21.5
21.4
21.5
21.7
66
21.9
21.8
21.8
21.6
21.7
80
21.9
21.5
21.8
21.5
21.6
21.8
Does the firing temperature affect the density of the bricks? Use α = 0.05. Find
the P-value of the test.
15. A shoe retailer conducted a study to determine whether there is a difference in
the number of pairs of shoes sold per day by stores according to the number of
competitors within a 1-mile radius and the location of the store. The company
researchers selected three types of stores for consideration in the study: standalone suburban stores, mall stores, and downtown stores. These stores vary in
the number of competing stores within a 1-mile radius, which have been reduced
to four categories: 0 competitors, 1 competitor, 2 competitors, and 3 or more
competitors. Suppose the following data represent the number of pairs of shoes
sold per day for each of these types of stores with the given number of
competitors. Use α = 0.05 and a two-way ANOVA to analyze the data.
Store
Location
0
41
30
45
25
22
31
18
29
33
StandAlone
Mall
Downtown
Number of Competitors
1
2
38
59
31
48
39
51
29
44
35
48
30
50
22
29
17
28
25
26
3 or more
47
40
39
43
42
53
24
27
32
16. To ascertain the stability of vitamin C in reconstituted frozen orange juice
concentrate stored in a refrigerator of a period of up to one week, the study
Vitamin C Retention in Reconstituted Frozen Orange Juice was conducted by the
Department of Human Nutrition and Foods at the Virginia Polytechnic Institute
and State University. Three types of frozen orange juice concentrate were tested
using 3 different time periods. The time periods refer to the number of days from
when the orange juice was blended until it was tested. The results, in milligrams
of ascorbic acid per liter, were recorded. Use a 0.05 level of significance to test
the hypothesis that
a. There is no difference in ascorbic acid contents among the different
brands of orange juice concentrate.
b. There is no difference in ascorbic acid contents for the different time
periods.
c. The brands of orange juice concentrate and the number of days from the
time the juice was blended until it is tested do not interact.
Brand
Richfood
Sealed
Sweet
Minute
Maid
0
52.6
49.8
56.0
49.6
52.5
51.8
54.2
46.5
48.0
48.4
52.0
53.6
Time (days)
3
49.4
49.2
42.8
53.2
48.8
44.0
44.0
42.4
48.0
47.0
48.2
49.6
7
42.7
40.4
49.2
42.0
48.5
45.2
48.8
47.6
44.0
43.2
43.3
47.6