· 8.6 A large laboratory has four types of devices used to determi.docx

· 8.6 A large laboratory has four types of devices used to
determine the pH of soil samples. The laboratory wants to
determine whether there are differences in the average readings
given by these devices. The lab uses 24 soil samples having
known pH in the study, and randomly assigns six of the samples
to each device. The soil samples are tested and the response
recorded is the difference between the pH reading of the device
and the known pH of the soil. These values, along with
summary statistics, are given in the following table
Devise
Sample
Response
A
1
-0.307
A
2
-0.294
A
3
0.079
A
4
0.019
A
5
-0.136
A
6
-0.324
B
1
-0.176
B

2
0.125
B
3
-0.013
B
4
0.082
B
5
0.091
B
6
0.459
C
1
0.137
C
2
-0.063
C
3
0.24
C
4
-0.05
C
5
0.318
C
6
0.154
D
1
-0.042
D

2
0.69
D
3
0.201
D
4
0.166
D
5
0.219
D
6
0.407
a. Based on your intuition, is there evidence to indicate any
difference among the mean differences in pH readings for the
four devices?
b. Run an analysis of variance to confirm or reject your
conclusion of part (a). Use a = .05
c. Compute the p-value of the F test in part (b).
d. What conditions must be satisfied for your analysis in parts
(b) and (c) to be valid?
e. Suppose the 24 soil samples have widely different pH values.
What problems may occur by simply randomly assigning the
soil samples to the different devices?
· 8.7 - It is conjectured that when fields are overgrazed by cattle
there will be a substantial reduction in the available grass
during the subsequent grazing season due to the compaction of
the soil. A horticulturist at the state agricultural experiment
station designs a study to evaluate the conjecture. Twenty-one
plots of land of nearly the same soil texture and suitable for
grazing are selected for the study. Three grazing regimens
selected for evaluation are randomly assigned to 7 plots each.

After the 21 plots are subjected to the grazing regimens for four
months, the researcher randomly selects 10 soil cores from each
plot and measures the bulk density (g/cm3) in each soil core.
The mean soil density of the 10 cores from each plot is given in
the following table.
1
2
3
5
19
25
17
10
15
12
9
12
10
7
9
4
5
8
a. Do the grazing regimens appear to yield different degrees of
effect on the amount of compacting in the soil? Justify your
answer using an α = .05 test.
b. Provide the level of significance of your test.
c. Do any of the conditions necessary for conducting your test
appear to be violated? Justify your answer.

8.18 – Refer to Example 7.8. The consumer testing agency was
interested in evaluating whether there was a difference in the
mean percentage increase in mpg of the three additives. In
Example 7.9, we showed that the data did not appear to have a
normal distribution.
a. Apply the natural logarithm transformation to the data. Do
the conditions for applying the AOV F test appear to hold for
the transformed data?
b. Test for a difference in the means of the three additives using
a = .05.
Subject
A1
A2

A3
1
3.0
1.8
1.3
2
1.2
6.3
12.6
3
1.0
5.2
10.0
4
0.7
3.7
10.5
5
1.1
5.4
10.8
6
0.6
2.9
5.9
7
1.2
6.0
12.1
8
0.1
0.3
0.6
9
0.7
3.6

18.6
10
1.9
9.3
18.7
11
0.6
2.8
5.5
12
0.0
0.0
0.0
13
1.6
8.1
18.2
14
4.0
19.9
22.3
15
0.1
0.3
0.6
· 8.20 – The use of computers as an instructional aid is widely
advocated as a means to capture the attention of the current
computer literate generation of students. A study was designed
to assess the effectiveness of using computers as a supplement
to the standard mode of instruction. Forty students in an
alternative school were randomly assigned to one of four
methods of teaching basic math skills. The four methods were
lectures only (L), lectures with remedial text book assistance
(L/R), lectures with computer assistance (L/C), and computer

instruction only (C). After a 10-week instructional period, an
exam evaluating basic math skills was taken by the students.
The difference in the scores on this exam and on an exam given
just prior to the 10-week instructional period for each student is
given in the following table. A few of the students did not
complete the program, thus producing an unequal number of
students in the four modes of instruction. The researchers want
to determine which method of instruction produces the largest
increase in test scores.
Method
L
L/R
L/C
C
1
9
5
9
17
2
2
2
12
12
3
2
3
2
26
4
6
11
17
1
5
16

16
12
47
6
11
11
20
27
7
9
3
20
28
8
0
31
10
9
4
21
20
10
2
(a) Which method of instruction appears to produce the largest
gain in scores?
(b) Is there significant evidence of a difference in the mean
gains for the four methods of instruction?
(c) Do the conditions for conducting statistical tests appear to
be satisfied? Justify your conclusions with graphs/tests.
(d) What is the target population for this study?

(e) Do the data collected in this study allow inferences to the
target population?
· 8.28 – The agricultural Experiment Station of a university
tested two different herbicides and their effects on crop yield.
From 90 acres set aside for the experiment, the station used
herbicide 1 on a random sample of 30 acres and herbicides 2 on
a second random sample of 30 acres; they used the remaining 30
acres as a control. At the end of the growing season, the yields
(in bushels per acre) were as follows:
Herbc1
Herbc2
Control
81.2
94.8
94.7
81.1
90.9
79.5
79.9
85.2
91.4
84.6
83.3
82.6
80.4
95.5
96.9
74.4
85.4

85.4
81.7
87.1
80.8
90.1
89.6
90.6
102.4
83.7
88.6
89.2
88.7
80.8
92.0
91.4
78.1
91.7
85.4
82.5
75.9
89.0
93.5
95.1
92.4
83.1
76.1
85.0
90.5
83.0
91.0
89.2
88.0
89.2
82.0
80.5
100.9

84.1
73.6
88.5
90.1
80.4
90.3
84.5
103.2
87.6
81.2
85.9
80.7
92.4
73.6
90.0
90.5
80.0
101.0
82.0
82.4
92.1
106.6
79.5
97.9
96.9
99.8
92.5
76.1
96.6
88.8
101.8
81.3
89.4
77.5
94.7
100.1

88.8
a. Use these data to conduct a one-way analysis of variance to
test whether there is a difference in the mean yields. Use α =
0.05
b. Construct 95% confidence intervals on the mean yields
c. Which of the mean yields appear to be different from the
control?
· 8.31 - Salary disputes and their eventual resolutions often
leave both employers and employees embittered by the entire
ordeal. To assess employee reactions to a recently devised
salary and fringe benefits plan, the H.R. department obtained
random samples of 15 employees from each of three divisions in
the company: Manufacturing, Marketing, and Research. The
personnel staff asked each employee sampled to respond (in
confidence) to a series of questions. Several employees refused
to cooperate, as reflected in the unequal sample sizes. The data
are given here:
Manuf
Market
Research
18.79
27.63
26.64
22.46
31.22
28.90
31.99
35.33
32.05
24.74
31.06
26.54
29.52
36.50
27.12

20.25
29.92
35.78
31.64
33.18
26.28
28.66
37.03
31.90
27.97
35.22
25.70
28.19
37.89
25.44
20.22
23.01
33.41
29.18
37.81
33.41
29.36
The data given above are the average responses from the
employees, with larger scores reflecting a higher degree of
satisfaction with management.
a. Write a model for this situation. Make sure to identify all the
terms in your model
b. Based on the summary of the scored responses, is there
significant evidence of a difference among the three divisions
with respect to their levels of satisfaction with management?

· 8.32 – Researchers record the yields of corn, in bushels per
plot, for four different varieties of corn, A, B, C, and D. In a
controlled greenhouse experiment, the researchers randomly
assign each variety to eight of 32 plots available for the study.
The yields are listed here.
A
B
C
D
2.5
3.6
4.3
2.8
3.6
3.9
4.4
2.9
2.8
4.1
4.5
3.1
2.7
4.3
4.1
2.4
3.1
2.9
3.5
3.2
3.4
3.5
3.4
2.5
2.9
3.8
3.2

3.6
3.5
3.7
4.6
2.7
a. Write an appropriate statistical model
b. Perform an analysis of variance on these data, and draw your
conclusions. Use α=.05
· 8.42 – A small corporation makes insulation shields for
electrical wires using three different types of machines. The
corporation wants to evaluate the variation in the inside
diameter dimensions of the shields produced by the machines. A
quality engineer at the corporation randomly selects shields
produced by each of the machines and records the inside
diameter of each shield (in millimeters). She wants to determine
whether the means and standard deviations of the three
machines differ.
Shield
Machine A
Machine B
Machine C
1
18.1
8.7
29.7
2
2.4
56.8
18.7
3
2.7
4.4
16.5
4
7.5

8.3
63.7
5
11.0
5.8
18.9
6
107.2
7
19.7
8
93.4
9
21.6
10
17.8
a. Conduct a test for the homogeneity of the population
variances. Use α = .05.
b. Would it be appropriate to proceed with an analysis of
variance based on the results of this test? Explain.
c. If the variances of the diameters are different, suggest a
transformation that may alleviate their differences, and then
conduct an analysis of variance to determine whether the mean
diameters differ. Use α = .05.
d. Compare the results of your analysis in part (c) to an analysis

of variance on the original diameters.
e. How could the engineer have designed her experiment
differently if she knew that the variances of machine B and
machine C were so much larger than that of machine A?

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