Questions:
- (45 points) A manufacturer of television sets is interested in the effect on tube conductivity of five different types of coating for color picture tubes. A completely randomized experiment is conducted and the following conductivity data are obtained:
Coating Type Conductivity
1 129 133 132 127 127
2 135 126 133 133 132
3 133 136 131 129 132
4 144 138 145 136 145
5 138 142 143 146 143
(10 points) Is there a difference in conductivity due to coating type? Conduct a test using α=0.05.
(10 points) Estimate the overall mean and the treatment effects.
(10 points) Test all pairs of means using the fisher LSD method with α=0.05
(5 points) Test all pairs of means using the Tukey's method with α=0.05.
(5 points) Check model adequacy.
(5 points) What are your recommendations to the manufacturer? We wish to minimize conductivity. - (30 points) A plant biologist conducted an experiment to compare the yields of 4 varieties of
peanuts (A, B, C, D). A plot of land was divided into 16 subplots (4 rows and 4 columns), where rows correspond to different watering frequencies and columns correspond to different soil types. The following Latin square design was run. The responses are given in the following table.
Watering Frequency Scheme Soil type
1 2 3 4
1 C=39 D=32 A=40 B=34
2 B=40 C=34 D=34 A=35
3 A=42 B=38 C=37 D=34
4 D=38 A=39 B=41 C=36
(5 points) There are three factors in this design. Which one(s) are nuisance factor and which one(s) are treatment factor? Why?
(15 points) Analyze the data and draw appropriate conclusions α=0.05.
(5 points) Check model adequacy.
(5 points) What are your recommendations to the plant biologist? - (40 points) An engineer is interested in the effects of cutting speed (A), tool geometry (B),
and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are
chosen, and four replicates of a 23 factorial design are run. The results are as follows.
A B C Treatment Combination Replicates
I II III IV
- - - (1) 30.82 26.79 23.98 30.92
- - - a 28.89 28.71 27.77 33.91
- + - b 8.22 39.58 16.96 33.09
- + - ab 38.08 38.21 23.92 31.13
- - + c 23.95 31.06 27.83 33.20
- - + ac 29.06 24.23 30.38 30.36
- + + bc 17.12 23.74 30.70 14.20
- + + abc 34.29 35.88 39.65 34.21
(10 points) Estimate the factor effects? Which effects appear to be large?
(10 points) Use the analysis of variance to confirm your conclusions for part (1). (α=0.05)
(10 points) Write down a regression model for predicting tool life (in hours) based on the results of the experiment. Based on the regression model, factor levels of A, B, and C would you recommend using?
(5 points) On the basis an analysis of the main effect and interaction plots, what coded
factor levels of A, B, and C would you recommend using?
(5 points) Analyze the residuals from the experiment. Are there any obvious problems?
- (45 points) An experimenter created a 7-factor 8-run design by assigning I = ABD = ACE=BCF=ABCG (or equivalently, D = AB, E = AC, F=BC, and G=ABC). The results are listed in Table 4.1.
(10 points) Is it a 2^k factorial design or fractional factorial design? If it is a factorial design, what is the k? If it is a fractional design, what is the fraction and resolution?
(10 points) What is the design generator? What is the alias structure?
(10 points) Analyze the data. Comment your findings.
(10 points) The fold-over runs appear in Table 4.2. Analyze the whole data (Table 4.1 + Table 4.2) and compare the analysis to your analysis in part (3). Are there any remaining uncertainties in your analysis of this experiment?
(5 points) What coded levels of the important factors would you recommend using, assume we want to maximize response?
Table 4.1
A B C D E F G Response
- - - + + + - 85.6
- - - - - + + 75.0
- + - - + - + 93.3
- + - + - - - 145.5
- - + + - - + 83.6
- - + - + - - 77.5
- + + - - + - 95.1
- + + + + + + 141.9
Table 4.2
A B C D E F G Response
- + + - - - + 91.4
- + + + + - - 136.7
- - + + - + - 82.4
- - + - + + + 73.4
- + - - + + - 94.2
- + - + - + + 143.9
- - - + + - + 87.4
- - - - - - - 71.8