STAT 3515Q: Design of Experiments — Spring 2025
Homework 3
Due date: March 21 (Friday), 11:59 p.m.
● This homework covers Chapter 4 and part of Chapter 5.
● The datasets are provided in the Excel file.
● The computational questions may be completed using SAS or R. However, for exam purposes, you should also understand how to solve them manually.
● Even if software is used for calculation, you still need to clearly write down the formulas used by the software and reasons for any conclusion.
● For model adequacy checking, you need to check the following plots (see Section 4.1.2, Figures 4.4-4.6): normal Q-Q plots; plot of residuals vs. fitted values; plots of residuals vs. each predictor (including residuals vs. treatments, residuals vs. blocking factor)
1. The effect of three different lubricating oils on fuel economy in diesel truck engines is being studied. Fuel economy is measured using brake-specific fuel consumption after the engine has been running for 15 minutes. Five different truck engines are available for the study, and the experimenters conduct the following RCBD.
|
|
Truck
|
|
Oil
|
1
|
2
|
3
|
4
|
5
|
|
1
|
0.500
|
0.634
|
0.487
|
0.329
|
0.512
|
|
2
|
0.535
|
0.675
|
0.520
|
0.435
|
0.540
|
|
3
|
0.513
|
0.595
|
0.488
|
0.400
|
0.510
|
(a) Discuss why RCBD is needed for this experiment and describe how the randomization is conducted. What is the difference between RCBD and the completely randomized design?
(b) Set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.
(c) Show the formula for the test statistic and compute its value.
(d) What is distribution of the test statistic under the null hypothesis?
(e) Using α = 0.05, what is your conclusion?
(f) Use the Tukey’s method with α = 0.05 to make comparisons among the three lubricating oils to determine specifically which oils differ in brake-specific fuel consumption.
(g) Conduct model adequacy checking.
2. The effect of five different ingredients (A, B, C, D, E) on the reaction time of a chemical process is being studied. Each batch of new material is only large enough to permit five runs to be made. Furthermore, each run requires approximately 1.5 hours, so only five runs can be made in one day. The experimenter decides to run the experiment as a Latin square so that day and batch effects may be systematically controlled. She obtains the data that follow.
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|
Day
|
|
Batch
|
1
|
2
|
3
|
4
|
5
|
|
1
|
A=8
|
B=7
|
D=1
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C=7
|
E=3
|
|
2
|
C=11
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E=2
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A=7
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D=3
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B=8
|
|
3
|
B=4
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A=9
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C=10
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E=1
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D=5
|
|
4
|
D=6
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C=8
|
E=6
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B=6
|
A=10
|
|
5
|
E=4
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D=2
|
B=3
|
A=8
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C=8
|
(a) Discuss why a Latin square is needed for this experiment. What are the features of a Latin square?
(b) Set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.
(c) Show the formula for the test statistic and compute its value.
(d) What is distribution of the test statistic under the null hypothesis?
(e) Using α = 0.05, what is your conclusion?
(f) Conduct model adequacy checking.
3. An industrial engineer is investigating the effect of four assembly methods (A, B, C, D) on the assembly time for a color television component. Four operators are selected for the study. Furthermore, the engineer knows that each assembly method produces such fatigue that the time required for the last assembly may be greater than the time required for the first, regardless of the method. That is, a trend develops in the required assembly time. Hence, a third factor, the order of assembly, is introduced. Moreover, the engineer suspects that the workplaces used by the four operators may represent an additional source of variation, so a fourth factor, workplace (α,β,γ,δ) is further introduced. This yields the Graeco-Latin square that follows.
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|
Operator
|
|
Order of Assembly
|
1
|
2
|
3
|
4
|
|
1
|
Cβ = 11
|
Bγ = 10
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Dδ = 14
|
Aα = 8
|
|
2
|
Bα = 8
|
Cδ = 12
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Aγ = 10
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Dβ = 12
|
|
3
|
Aδ = 9
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Dα = 11
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Bβ = 7
|
Cγ = 15
|
|
4
|
Dγ = 9
|
Aβ = 8
|
Cα = 18
|
Bδ = 6
|
(a) Discuss why a Graeco-Latin square is needed for this experiment. What are the features of a Graeco-Latin square?
(b) Set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.
(c) Show the formula for the test statistic and compute its value.
(d) What is distribution of the test statistic under the null hypothesis?
(e) Using α = 0.05, what is your conclusion?
(f) Conduct model adequacy checking.
4. An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. He selects three feed rates and four depths of cut. He then conducts a factorial experiment and obtains the following data:
|
Feed Rate
(in/min)
|
Depth of Cut (in)
|
|
0.15 0.18 0.2 0.25
|
|
0.2
|
74 79 82 99
64 68 88 104
60 73 92 96
|
|
0.25
|
92 98 99 104
86 104 108 110
88 88 95 99
|
|
0.3
|
99 104 108 114
98 99 110 111
102 95 99 107
|
(a) Briefly describe how to conduct the randomization for this design.
(b) Specify the statistical model and the corresponding assumptions (including constraints). Then set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.
(c) Show the formula for the test statistics and compute their values.
(d) What are distributions of the test statistics under the null hypothesis?
(e) Using α = 0.05, what is your conclusion?
(f) Obtain parameter estimates for the fitted model.
(g) Use the Tukey’s method with α = 0.05 to make comparisons among different feed rates and draw conclusions.
(h) Use the Tukey’s method with α = 0.05 to make comparisons among different depths of cut and draw conclusions.
(i) Do we need to perform slicing? If so, conduct the analysis and draw conclusions.
(j) Conduct model adequacy checking.