Download Exam 2 with Solution for Experimental Design and Analysis | 22S 158 and more Exams Statistics in PDF only on Docsity!
NOTE: in 1c, Professor Lenth forgot that this was an unbalanced design making this
a much harder problem than he thought. He accepted answers such that
c
1 i
c
2 i
10 pts A. Explain the steps needed to assess the significance of nonadditivity when using
the Tukey 1 d.f. test.
5 pts B. If you proceeded with your above steps after fitting a model that included both
two-way interactions and three-way interaction, would this test for
nonadditivity make sense? Explain.
22S:158/165 Design: Exam 2 April 2, 2008 Corrected solutions
Show your work. Calculator and 2 pages of crib notes are allowed. Some statistical tables are included at the end of the exam.
Note: Students in 22S:158 should do problems 1–5; students in 22S:165 should do all 6 problems. This exam counts 60% of the
total exam grade, and the take home is 40%. The length of the exam is 50 minutes. It is up to you to keep track of your available
time, pace yourself, and decide what order to work on the problems. Write your answers directly on the paper.
1. We have the following data from tests of wind resistance of cars of 5 different models.
Model Impala Highlander Escalade Caliber Focus
Type Sedan SUV SUV Sedan Sedan
n
i
y ¯
i·
5 pts A. Give a set of contrast coefficients that is useful for comparing the sedans with the SUVs.
Solution: w = {
1
3
1
2
1
2
1
3
1
3
} or any nonzero multiple thereof
10 pts B. From the ANOVA table, we find that MS
E
= 2.78 (you need to figure out the d.f.). Calculate a 95%
confidence interval for the above contrast.
Solution:
The observed contrast is (3.57 + 4.31 + 3.46)/3 − (4.23 + 4.64)/2 = −.655. Its standard error is
MS
E ∑
w
2
i
/n
i
2
2
There are N = 19 observations so there are N − g = 14 d.f.; the 95% CI for the same contrast of the true
means means is
−.655 ± 2.14 × .8267 = −.655 ± 1.769 → (−2.42, 1.11)
5 pts C. Find another set of contrast coefficients that is orthogonal to the ones you answered in (A).
Solution: v = {0, 2.4, −2.4, 1, − 1 } or any other v such that ∑
w
i
v
i
/n
i
= 0. (I overlooked the unequal ns
when I wrote this problem, making it much harder; I’ll accept answers for which ∑
w
i
v
i
2. You have data from a factorial experiment involving three factors A, B, C. Answer these questions
concerning nonadditivity tests.
10 pts A. Explain the steps needed to assess the significance of nonadditivity, relative to the model
y
ijkl
= μ + α
i
j
k
ijkl
Solution:
- Fit this model to the data and obtain the fitted values yˆ
i
- Calculate a new variable Nonadd = yˆ
2
i
- Fit the model with Nonadd as an additional predictor. Its t statistic is a test of nonadditivity
(equivalently, the F statistic comparing these two models)
5 pts B. Would a nonadditivity test make sense if the model includes all main effects and two- and three-way
interactions? Explain.
Solution: No. The model will fit the cell means perfectly, so there can be no curvature in the residuals versus
fitted plot; and nonadditivity tests this curvature.
22S:158/165 Design: Exam 2 April 2, 2008 Corrected solutions
Show your work. Calculator and 2 pages of crib notes are allowed. Some statistical tables are included at the end of the exam.
Note: Students in 22S:158 should do problems 1–5; students in 22S:165 should do all 6 problems. This exam counts 60% of the
total exam grade, and the take home is 40%. The length of the exam is 50 minutes. It is up to you to keep track of your available
time, pace yourself, and decide what order to work on the problems. Write your answers directly on the paper.
1. We have the following data from tests of wind resistance of cars of 5 different models.
Model Impala Highlander Escalade Caliber Focus
Type Sedan SUV SUV Sedan Sedan
n
i
y ¯
i·
5 pts A. Give a set of contrast coefficients that is useful for comparing the sedans with the SUVs.
Solution: w = {
1
3
1
2
1
2
1
3
1
3
} or any nonzero multiple thereof
10 pts B. From the ANOVA table, we find that MS
E
= 2.78 (you need to figure out the d.f.). Calculate a 95%
confidence interval for the above contrast.
Solution:
The observed contrast is (3.57 + 4.31 + 3.46)/3 − (4.23 + 4.64)/2 = −.655. Its standard error is
MS
E ∑
w
2
i
/n
i
2
2
There are N = 19 observations so there are N − g = 14 d.f.; the 95% CI for the same contrast of the true
means means is
−.655 ± 2.14 × .8267 = −.655 ± 1.769 → (−2.42, 1.11)
5 pts C. Find another set of contrast coefficients that is orthogonal to the ones you answered in (A).
Solution: v = {0, 2.4, −2.4, 1, − 1 } or any other v such that
w
i
v
i
/n
i
= 0. (I overlooked the unequal ns
when I wrote this problem, making it much harder; I’ll accept answers for which ∑
w
i
v
i
2. You have data from a factorial experiment involving three factors A, B, C. Answer these questions
concerning nonadditivity tests.
10 pts A. Explain the steps needed to assess the significance of nonadditivity, relative to the model
y
ijkl
i
j
k
ijkl
Solution:
- Fit this model to the data and obtain the fitted values yˆ
i
2. Calculate a new variable Nonadd =
y
2
i
- Fit the model with Nonadd as an additional predictor. Its t statistic is a test of nonadditivity
(equivalently, the F statistic comparing these two models)
5 pts B. Would a nonadditivity test make sense if the model includes all main effects and two- and three-way
interactions? Explain.
Solution: No. The model will fit the cell means perfectly, so there can be no curvature in the residuals versus
fitted plot; and nonadditivity tests this curvature.
1. We have the following data from tests of wind resistance of cars of 5 different models.
Model Impala Highlander Escalade Caliber Focus
Type Sedan SUV SUV Sedan Sedan
n
i
y ¯
i·
5 pts A. Give a set of contrast coefficients that is useful for comparing the sedans with the SUVs.
Solution: w = {
1
3
1
2
1
2
1
3
1
3
} or any nonzero multiple thereof
10 pts B. From the ANOVA table, we find that MS
E
= 2.78 (you need to figure out the d.f.). Calculate a 9
confidence interval for the above contrast.
Solution:
The observed contrast is (3.57 + 4.31 + 3.46)/3 − (4.23 + 4.64)/2 = −.655. Its standard error is
MS
E ∑
w
2
i
/n
i
2
2
There are N = 19 observations so there are N − g = 14 d.f.; the 95% CI for the same contrast of t
means means is
−.655 ± 2.14 × .8267 = −.655 ± 1.769 → (−2.42, 1.11)
5 pts C. Find another set of contrast coefficients that is orthogonal to the ones you answered in (A).
Solution: v = {0, 2.4, −2.4, 1, − 1 } or any other v such that
w
i
v
i
/n
i
= 0. (I overlooked the uneq
when I wrote this problem, making it much harder; I’ll accept answers for which ∑ w
i
v
i
2. You have data from a factorial experiment involving three factors A, B, C. Answer these questions
concerning nonadditivity tests.
10 pts A. Explain the steps needed to assess the significance of nonadditivity, relative to the model
y
ijkl
i
j
k
ijkl
Solution:
1. Fit this model to the data and obtain the fitted values
y
i
2. Calculate a new variable Nonadd = yˆ
2
i
3. Fit the model with Nonadd as an additional predictor. Its t statistic is a test of nonadditivity
(equivalently, the F statistic comparing these two models)
5 pts B. Would a nonadditivity test make sense if the model includes all main effects and two- and thre
interactions? Explain.
Solution: No. The model will fit the cell means perfectly, so there can be no curvature in the residu
fitted plot; and nonadditivity tests this curvature.
Show your work. Calculator and 2 pages of crib notes are allowed. Some statistical tables are included at the end of the ex
Note: Students in 22S:158 should do problems 1–5; students in 22S:165 should do all 6 problems. This exam counts 60%
total exam grade, and the take home is 40%. The length of the exam is 50 minutes. It is up to you to keep track of your a
time, pace yourself, and decide what order to work on the problems. Write your answers directly on the paper.
1. We have the following data from tests of wind resistance of cars of 5 different models.
Model Impala Highlander Escalade Caliber Focus
Type Sedan SUV SUV Sedan Sedan
n
i
y
i·
5 pts A. Give a set of contrast coefficients that is useful for comparing the sedans with the SUVs.
Solution: w = {
1
3
1
2
1
2
1
3
1
3
} or any nonzero multiple thereof
10 pts B. From the ANOVA table, we find that MS
E
= 2.78 (you need to figure out the d.f.). Calculate a 95%
confidence interval for the above contrast.
Solution:
The observed contrast is (3.57 + 4.31 + 3.46)/3 − (4.23 + 4.64)/2 = −.655. Its standard error is
MS
E ∑
w
2
i
/n
i
2
2
There are N = 19 observations so there are N − g = 14 d.f.; the 95% CI for the same contrast of the
means means is
−.655 ± 2.14 × .8267 = −.655 ± 1.769 → (−2.42, 1.11)
5 pts C. Find another set of contrast coefficients that is orthogonal to the ones you answered in (A).
Solution: v = {0, 2.4, −2.4, 1, − 1 } or any other v such that
w
i
v
i
/n
i
= 0. (I overlooked the unequa
when I wrote this problem, making it much harder; I’ll accept answers for which
w
i
v
i
2. You have data from a factorial experiment involving three factors A, B, C. Answer these questions
concerning nonadditivity tests.
10 pts A. Explain the steps needed to assess the significance of nonadditivity, relative to the model
y
ijkl
i
j
k
ijkl
Solution:
1. Fit this model to the data and obtain the fitted values yˆ
i
2. Calculate a new variable Nonadd = yˆ
2
i
3. Fit the model with Nonadd as an additional predictor. Its t statistic is a test of nonadditivity
(equivalently, the F statistic comparing these two models)
5 pts B. Would a nonadditivity test make sense if the model includes all main effects and two- and three-w
interactions? Explain.
Solution: No. The model will fit the cell means perfectly, so there can be no curvature in the residuals
fitted plot; and nonadditivity tests this curvature.
SNK HSD LSD REGWR Scheffe Bonferroni (none)
In the context of testing all pairwise comparisons of means…
5 pts A. Which one(s) controls the strong FWER, but is most conservative?
5 pts B. Which one(s) does not control the FWER?
5 pts C. Assuming you have equal sample sizes, which of the methods use the same
critical value for testing significance for all pairs? ( we don’t mean the different
methods are using the same critical value, but we mean WITHIN a method).
5pts D. In which one(s) is it possible that a comparison is not even tested due to some
other test being nonsignificant?
22S:158/165 Design: Exam 2
3. The answers to each part of this question should be chosen from this list:
SNK HSD LSD REGWR Protected LSD Scheffé BSD (none)
In the context of testing all pairwise comparisons of means,
5 pts A. Which one(s) are conservative in terms of strongly protecting the familywise error rate?
Solution: Scheffé, BSD
5 pts B. Which one(s) do not protect the familywise error rate?
Solution: LSD
5 pts C. Which one(s) use the same critical value for every comparison (assuming the sample sizes are equal)?
Solution: HSD, LSD, Protected LSD, Scheffé, BSD
5 pts D. In which one(s) is it possible that a comparison is not even tested due to some other test being
nonsignificant?
Solution: SNK, REGWR, Protected LSD
10 pts 4. We have completed an internet-marketing experiment involving two factors: P: promotional material used
for the product (2 levels), and A: number of advertisements sent to potential customers (5 levels: 1 through
5). The response is the number of orders received, in 4 independent campaigns with each factor combination
We do the analysis with the two-factor model, and find a significant interaction between P and A. The
diagnostic plots all look OK. Briefly describe the steps you would take to complete the analysis.
Solution:
First, I’d construct an interaction plot. Since A is quantitative, I’d plot mean response versus A, with separate
curves for each P. Second, again since A is quantitative, I’d find a good regression model that fits a separate tren
in A (a line or a curve) separately for each P. The lowest degree polynomial that fits well wouild be a nice
parsimonious way to characterize the results. Note that we need separate trends because the interaction is
significant.
5. Suppose we collect 10 observations on each of three treatments; the actual treatment means are μ
1
2
= 11, and μ
3
= 14; and the population standard deviation is σ = 6. All other things being equal, will the
power of the ANOVA F test increase , decrease , or stay the same if...
5 pts A.... we increase the significance level, E?
Solution: Power will increase; Increasing E makes it easier to reject H
0
5 pts B.... the means are instead μ
1
2
= 18, and μ
3
Solution: power will decrease; these means are less variable than the original ones
5 pts C.... the value of σ is decreased to 4.
Solution: Power will increase, because there is less error variation making for clearer conclusions.
10 pts 6. Answer this question only if you are enrolled in 22S:165 (graduate statistics major)
Refer to the situation in Problem 5. Using the values of μ
i
and σ stated in the introduction to the problem,
what is the distribution of F? Give the name of the distribution and numerical values of all of its parameters.
Solution:
22S:158/165 Design: Exam 2 2
3. The answers to each part of this question should be chosen from this list:
SNK HSD LSD REGWR Protected LSD Scheffé BSD (none)
In the context of testing all pairwise comparisons of means,
5 pts A. Which one(s) are conservative in terms of strongly protecting the familywise error rate?
Solution: Scheffé, BSD
5 pts B. Which one(s) do not protect the familywise error rate?
Solution: LSD
5 pts C. Which one(s) use the same critical value for every comparison (assuming the sample sizes are equal)?
Solution: HSD, LSD, Protected LSD, Scheffé, BSD
5 pts D. In which one(s) is it possible that a comparison is not even tested due to some other test being
nonsignificant?
Solution: SNK, REGWR, Protected LSD
10 pts 4. We have completed an internet-marketing experiment involving two factors: P: promotional material used
for the product (2 levels), and A: number of advertisements sent to potential customers (5 levels: 1 through
5). The response is the number of orders received, in 4 independent campaigns with each factor combination.
We do the analysis with the two-factor model, and find a significant interaction between P and A. The
diagnostic plots all look OK. Briefly describe the steps you would take to complete the analysis.
Solution:
First, I’d construct an interaction plot. Since A is quantitative, I’d plot mean response versus A, with separate
curves for each P. Second, again since A is quantitative, I’d find a good regression model that fits a separate trend
in A (a line or a curve) separately for each P. The lowest degree polynomial that fits well wouild be a nice
parsimonious way to characterize the results. Note that we need separate trends because the interaction is
significant.
5. Suppose we collect 10 observations on each of three treatments; the actual treatment means are μ
1
μ 2
= 11, and μ 3
= 14; and the population standard deviation is σ = 6. All other things being equal, will the
power of the ANOVA F test increase , decrease , or stay the same if...
5 pts A.... we increase the significance level, E?
Solution: Power will increase; Increasing E makes it easier to reject H
0
5 pts B.... the means are instead μ
1
= 20, μ
2
= 18, and μ
3
Solution: power will decrease; these means are less variable than the original ones
5 pts C.... the value of σ is decreased to 4.
Solution: Power will increase, because there is less error variation making for clearer conclusions.
10 pts 6. Answer this question only if you are enrolled in 22S:165 (graduate statistics major)
Refer to the situation in Problem 5. Using the values of μ
i
and σ stated in the introduction to the problem,
what is the distribution of F? Give the name of the distribution and numerical values of all of its parameters.
Solution:
F has a noncentral F distribution with (2, 27) degrees of freedom and noncentrality parameter ζ = ∑ n
i
α
2
i
/ σ
2
Since the average of the means is 12.333, ζ = 10 × (.
2
2
2
2
22S:158/165 Design: Exam 2 2
3. The answers to each part of this question should be chosen from this list:
SNK HSD LSD REGWR Protected LSD Scheffé BSD (none)
In the context of testing all pairwise comparisons of means,
5 pts A. Which one(s) are conservative in terms of strongly protecting the familywise error rate?
Solution: Scheffé, BSD
5 pts B. Which one(s) do not protect the familywise error rate?
Solution: LSD
5 pts C. Which one(s) use the same critical value for every comparison (assuming the sample sizes are equal)?
Solution: HSD, LSD, Protected LSD, Scheffé, BSD
5 pts D. In which one(s) is it possible that a comparison is not even tested due to some other test being
nonsignificant?
Solution: SNK, REGWR, Protected LSD
10 pts 4. We have completed an internet-marketing experiment involving two factors: P: promotional material used
for the product (2 levels), and A: number of advertisements sent to potential customers (5 levels: 1 through
5). The response is the number of orders received, in 4 independent campaigns with each factor combination.
We do the analysis with the two-factor model, and find a significant interaction between P and A. The
diagnostic plots all look OK. Briefly describe the steps you would take to complete the analysis.
Solution:
First, I’d construct an interaction plot. Since A is quantitative, I’d plot mean response versus A, with separate
curves for each P. Second, again since A is quantitative, I’d find a good regression model that fits a separate trend
in A (a line or a curve) separately for each P. The lowest degree polynomial that fits well wouild be a nice
parsimonious way to characterize the results. Note that we need separate trends because the interaction is
significant.
5. Suppose we collect 10 observations on each of three treatments; the actual treatment means are μ
1
μ
2
= 11, and μ
3
= 14; and the population standard deviation is σ = 6. All other things being equal, will the
power of the ANOVA F test increase , decrease , or stay the same if...
5 pts A.... we increase the significance level, E?
Solution: Power will increase; Increasing E makes it easier to reject H
0
5 pts B.... the means are instead μ
1
= 20, μ
2
= 18, and μ
3
Solution: power will decrease; these means are less variable than the original ones
5 pts C.... the value of σ is decreased to 4.
Solution: Power will increase, because there is less error variation making for clearer conclusions.
10 pts 6. Answer this question only if you are enrolled in 22S:165 (graduate statistics major)
Refer to the situation in Problem 5. Using the values of μ
i
and σ stated in the introduction to the problem,
what is the distribution of F? Give the name of the distribution and numerical values of all of its parameters.
Solution:
F has a noncentral F distribution with (2, 27) degrees of freedom and noncentrality parameter ζ = ∑ n
i
α
2
i
/ σ
2
Since the average of the means is 12.333, ζ = 10 × (.
2
2
2
2
22S:158/165 Design: Exam 2 2
3. The answers to each part of this question should be chosen from this list:
SNK HSD LSD REGWR Protected LSD Scheffé BSD (none)
In the context of testing all pairwise comparisons of means,
5 pts A. Which one(s) are conservative in terms of strongly protecting the familywise error rate?
Solution: Scheffé, BSD
5 pts B. Which one(s) do not protect the familywise error rate?
Solution: LSD
5 pts C. Which one(s) use the same critical value for every comparison (assuming the sample sizes are equal)?
Solution: HSD, LSD, Protected LSD, Scheffé, BSD
5 pts D. In which one(s) is it possible that a comparison is not even tested due to some other test being
nonsignificant?
Solution: SNK, REGWR, Protected LSD
10 pts 4. We have completed an internet-marketing experiment involving two factors: P: promotional material used
for the product (2 levels), and A: number of advertisements sent to potential customers (5 levels: 1 through
5). The response is the number of orders received, in 4 independent campaigns with each factor combination.
We do the analysis with the two-factor model, and find a significant interaction between P and A. The
diagnostic plots all look OK. Briefly describe the steps you would take to complete the analysis.
Solution:
First, I’d construct an interaction plot. Since A is quantitative, I’d plot mean response versus A, with separate
curves for each P. Second, again since A is quantitative, I’d find a good regression model that fits a separate trend
in A (a line or a curve) separately for each P. The lowest degree polynomial that fits well wouild be a nice
parsimonious way to characterize the results. Note that we need separate trends because the interaction is
significant.
5. Suppose we collect 10 observations on each of three treatments; the actual treatment means are μ
1
μ
2
= 11, and μ
3
= 14; and the population standard deviation is σ = 6. All other things being equal, will the
power of the ANOVA F test increase , decrease , or stay the same if...
5 pts A.... we increase the significance level, E?
Solution: Power will increase; Increasing E makes it easier to reject H
0
5 pts B.... the means are instead μ
1
= 20, μ
2
= 18, and μ
3
Solution: power will decrease; these means are less variable than the original ones
5 pts C.... the value of σ is decreased to 4.
Solution: Power will increase, because there is less error variation making for clearer conclusions.
10 pts 6. Answer this question only if you are enrolled in 22S:165 (graduate statistics major)
Refer to the situation in Problem 5. Using the values of μ
i
and σ stated in the introduction to the problem,
what is the distribution of F? Give the name of the distribution and numerical values of all of its parameters.
Solution:
F has a noncentral F distribution with (2, 27) degrees of freedom and noncentrality parameter ζ = ∑
n
i
α
2
i
/ σ
2
Since the average of the means is 12.333, ζ = 10 × (.
2
2
2
2
22S:158/165 Design: Exam 2 2
3. The answers to each part of this question should be chosen from this list:
SNK HSD LSD REGWR Protected LSD Scheffé BSD (none)
In the context of testing all pairwise comparisons of means,
5 pts A. Which one(s) are conservative in terms of strongly protecting the familywise error rate?
Solution: Scheffé, BSD
5 pts B. Which one(s) do not protect the familywise error rate?
Solution: LSD
5 pts C. Which one(s) use the same critical value for every comparison (assuming the sample sizes are equal)?
Solution: HSD, LSD, Protected LSD, Scheffé, BSD
5 pts D. In which one(s) is it possible that a comparison is not even tested due to some other test being
nonsignificant?
Solution: SNK, REGWR, Protected LSD
10 pts 4. We have completed an internet-marketing experiment involving two factors: P: promotional material used
for the product (2 levels), and A: number of advertisements sent to potential customers (5 levels: 1 through
5). The response is the number of orders received, in 4 independent campaigns with each factor combination.
We do the analysis with the two-factor model, and find a significant interaction between P and A. The
diagnostic plots all look OK. Briefly describe the steps you would take to complete the analysis.
Solution:
First, I’d construct an interaction plot. Since A is quantitative, I’d plot mean response versus A, with separate
curves for each P. Second, again since A is quantitative, I’d find a good regression model that fits a separate trend
in A (a line or a curve) separately for each P. The lowest degree polynomial that fits well wouild be a nice
parsimonious way to characterize the results. Note that we need separate trends because the interaction is
significant.
5. Suppose we collect 10 observations on each of three treatments; the actual treatment means are μ
1
μ
2
= 11, and μ
3
= 14; and the population standard deviation is σ = 6. All other things being equal, will the
power of the ANOVA F test increase , decrease , or stay the same if...
5 pts A.... we increase the significance level, E?
Solution: Power will increase; Increasing E makes it easier to reject H
0
5 pts B.... the means are instead μ
1
= 20, μ
2
= 18, and μ
3
Solution: power will decrease; these means are less variable than the original ones
5 pts C.... the value of σ is decreased to 4.
Solution: Power will increase, because there is less error variation making for clearer conclusions.
10 pts 6. Answer this question only if you are enrolled in 22S:165 (graduate statistics major)
Refer to the situation in Problem 5. Using the values of μ
i
and σ stated in the introduction to the problem,
what is the distribution of F? Give the name of the distribution and numerical values of all of its parameters.
Solution:
F has a noncentral F distribution with (2, 27) degrees of freedom and noncentrality parameter ζ = ∑
n
i
α
2
i
/ σ
2
Since the average of the means is 12.333, ζ = 10 × (.
2
2
2
2
22S:158/165 Design: Exam 2 2
3. The answers to each part of this question should be chosen from this list:
SNK HSD LSD REGWR Protected LSD Scheffé BSD (none)
In the context of testing all pairwise comparisons of means,
5 pts A. Which one(s) are conservative in terms of strongly protecting the familywise error rate?
Solution: Scheffé, BSD
5 pts B. Which one(s) do not protect the familywise error rate?
Solution: LSD
5 pts C. Which one(s) use the same critical value for every comparison (assuming the sample sizes are equal)?
Solution: HSD, LSD, Protected LSD, Scheffé, BSD
5 pts D. In which one(s) is it possible that a comparison is not even tested due to some other test being
nonsignificant?
Solution: SNK, REGWR, Protected LSD
10 pts 4. We have completed an internet-marketing experiment involving two factors: P: promotional material used
for the product (2 levels), and A: number of advertisements sent to potential customers (5 levels: 1 through
5). The response is the number of orders received, in 4 independent campaigns with each factor combination.
We do the analysis with the two-factor model, and find a significant interaction between P and A. The
diagnostic plots all look OK. Briefly describe the steps you would take to complete the analysis.
Solution:
First, I’d construct an interaction plot. Since A is quantitative, I’d plot mean response versus A, with separate
curves for each P. Second, again since A is quantitative, I’d find a good regression model that fits a separate trend
in A (a line or a curve) separately for each P. The lowest degree polynomial that fits well wouild be a nice
parsimonious way to characterize the results. Note that we need separate trends because the interaction is
significant.
5. Suppose we collect 10 observations on each of three treatments; the actual treatment means are μ
1
μ 2
= 11, and μ 3
= 14; and the population standard deviation is σ = 6. All other things being equal, will the
power of the ANOVA F test increase , decrease , or stay the same if...
5 pts A.... we increase the significance level, E?
Solution: Power will increase; Increasing E makes it easier to reject H
0
5 pts B.... the means are instead μ
1
= 20, μ 2
= 18, and μ 3
Solution: power will decrease; these means are less variable than the original ones
5 pts C.... the value of σ is decreased to 4.
Solution: Power will increase, because there is less error variation making for clearer conclusions.
10 pts 6. Answer this question only if you are enrolled in 22S:165 (graduate statistics major)
Refer to the situation in Problem 5. Using the values of μ
i
and σ stated in the introduction to the problem,
what is the distribution of F? Give the name of the distribution and numerical values of all of its parameters.
Solution:
F has a noncentral F distribution with (2, 27) degrees of freedom and noncentrality parameter ζ = ∑
n
i
α
2
i
/ σ
2
Since the average of the means is 12.333, ζ = 10 × (.
2
2
2
2
