DIRECTIONS: Please upload a single PDF document will your solutions, clearly indicating the problem numbers where each answer starts. Your answers must be in the order of the questions shown below. Unclear homework/scans or unsorted answers will be penalized. Show your work if you want full credit. Remember, long portions of computer output will not be graded and will receive no credit, you must cut and paste any relevant spreadsheet or Matlab output into your homework solutions. You may only use a spreadsheet software (e.g., MS Excel) for ANOVA computations (contrast tables, effects and sums of squares). MATLAB is allowed for matrix computations in regression analysis in problem #1 and for NPP of the effects (Daniels’ plots) and p-values for the problems where this is needed. You may not use any other functions from MATLAB other than for matrix multi- plications, NPPs and p-values in F tests. Likewise, when using a spreadsheet, you may not use functions/add-ins that do ANOVA or regression computations for you.
Academic integrity policy reminder: work is to be done strictly individually. You may not get help from anybody else within or outside your classmates group. See class syllabus for a fuller statement of this policy.
1. Consider a 23−1 experimental design where I = ABC and, since all factors are numeric (non-categorical), you wish to analyze the DOE data using regression techniques. Sup- pose the response values are a = 15, b = 10, c = 35, and abc = 25.
(a) Construct the X matrix for the regression model y = β0+β1x1+β2x2+ε. Find the least squares estimates of the parameters using the expression for βˆ in regression analysis.
(b) Construct the X matrix for the model y = β0 + β1x1 + β2x2 + β12x1x2 + β13x1x3 +β23x2x3 + ε. Can you find β^ using regression for this model? Why?
2. Suppose an engineer wishes to design a 24−1 experiment with I = ABCD (so D =ABC).
(a) Construct the experimental design. Give the table of ’pluses and minuses’ (i.e., table of contrasts) for the main effects. How many runs does this experiment have?
(b) Give the table of contrasts for all the possible 15 effects. How many uniquely different columns are there?
(c) Confirm from comparing the contrasts (columns) in part (b) above that main effects are confounded with a 3 factor interactions and that 2 factor interactions are confounded with another 2-factor interaction.
(d) What is the resolution of this design?
3. A car manufacturer is developing a paint which a customer required to have high glossiness and acceptable abrasion resistance with the glossiness response (y1) measures in a scale of 1 to 100 and abrasion resistance (y2) measured on a 1 to 10 scale. The paint technology group of the company decided to run a completely randomized 28−4 design with design generators E = ABC, F = ABD, G = ACD and H = BCD. The observed responses, listed using Yates’ notation were:
|
Treatment |
Glossiness y1 |
Abrasion resistance y2 |
|
(1) |
53 |
6.3 |
|
aefg |
60 |
6.1 |
|
befh |
68 |
5.5 |
|
abgh |
78 |
2.1 |
|
cegh |
48 |
6.9 |
|
acfh |
67 |
5.1 |
|
bcfg |
55 |
6.4 |
|
abce |
78 |
2.5 |
|
dfgh |
49 |
8.2 |
|
adeh |
68 |
3.1 |
|
bdeg |
61 |
4.3 |
|
abdf |
81 |
3.2 |
|
cdef |
52 |
7.1 |
|
acdg |
70 |
3.4 |
|
bcdh |
65 |
3.0 |
|
abcdefgh |
82 |
2.8 |
(a) Construct the table of “pluses and minuses” for the 8 main effects (which define the experimental design) and for all the 2-factor interactions. Then estimate the main effects and 2-factor interactions for the glossiness and abrasion resistance responses (note this is a fractional factorial, and there will be aliased effects).
(b) Do a normal probability plot of the effects for each of the two responses (glossiness, y1 and abrasion resistance, y2) separately and identify the effects that seem to be significant, considering the alias structure of the experiment. Then complete an ANOVA test for each response (y1 and y2) to formally test for the significance of the effects you detected in the NPP of the effects pooling into error the SS’s of effects that appear negligible. Report the significant effects and the standard
deviation (s = responses.σˆ) of the experimental noise associated with each of the two
(c) Assume that all 8 factors are numeric and were coded into the (-1,1) coding convention. Derive fitted regression models for y1 and y2 from the significant effects (include an intercept in each case).
(d) The customer desires glossiness (y1) to be as high as possible, and abrasion resis- tance (y2) to be at least equal to 5. Find the levels of the significant factors you found in parts b)-c). You may use the Excel “solver” maximizing yˆ1 subject to a constraint in yˆ2 (include also constraints to keep each factor in the range ( 1, 1)) or you can draw contour plots of your responses and eyeballing a solution from the plots. Compare with the best solution you could obtain by “picking the winner”, i.e., by picking the best run among the 16 tests in the experiment.
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