A nickel-titanium alloy is used to make components for jet turbine aircraft engi
ID: 3066545 • Letter: A
Question
A nickel-titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a potentially serious problem in the final part, as it can lead to non-recoverable failure. A test is run at the parts producer to determine the effects of four factors on cracks. The four factors are pouring temperature (A), titanium content (B), heat treatment method (C), and the amount of grain refiner used (D). Two replicated of a 24 design are run, and the length of crack (in ?m) induced in a sample coupon subjected to a standard test is measured. The data are shown below:
Treatment Replicate eplicate A B CD Combination 7.037 14.707 11.635 17.273 10.403 4.368 9.360 13.440 8.561 6.376 15.219 12.089 17.815 10.151 4.098 9.253 12.923 8.951 ac bc abc Estimate the factor effects. Which factors appear to be large? a. b. Conduct an analysis of variance. Do any of the factors affect cracking? Write down a regression model that can be used to predict crack length as a function of the significant main effects and interactions you have identified in part (b) c. d. Analyze the residuals from this experiment. e. Is there an indication that any of the factors affect the variability in cracking? f. What recommendations would you make regarding process operations? Use interaction and/or main effect plots to assist in drawing conclusionsExplanation / Answer
a.
Coefficients
Term Coef SE Coef T-Value P-Value VIF
Constant 11.9568 0.0622 192.10 0.000
A
-1 -1.4782 0.0622 -23.75 0.000 1.00
B
-1 -2.0192 0.0622 -32.44 0.000 1.00
C
-1 1.7669 0.0622 28.39 0.000 1.00
D
-1 -0.9476 0.0622 -15.22 0.000 1.00
A*B
-1 -1 0.9983 0.0622 16.04 0.000 1.00
A*C
-1 -1 -1.9726 0.0622 -31.69 0.000 1.00
A*D
-1 -1 0.0070 0.0622 0.11 0.912 1.00
B*C
-1 -1 0.0167 0.0622 0.27 0.791 1.00
B*D
-1 -1 0.0549 0.0622 0.88 0.391 1.00
C*D
-1 -1 -0.0072 0.0622 -0.12 0.910 1.00
A*B*C
-1 -1 -1 -1.5375 0.0622 -24.70 0.000 1.00
A*B*D
-1 -1 -1 -0.0802 0.0622 -1.29 0.216 1.00
A*C*D
-1 -1 -1 -0.0408 0.0622 -0.66 0.521 1.00
B*C*D
-1 -1 -1 0.0134 0.0622 0.22 0.832 1.00
A*B*C*D
-1 -1 -1 -1 -0.0242 0.0622 -0.39 0.703 1.00Titanium content has higher effect than other factor.
General Factorial Regression: Crack leng versus A, B, C, D
B.
Factor Information
Factor Levels Values
A 2 -1, 1
B 2 -1, 1
C 2 -1, 1
D 2 -1, 1
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Model 15 561.473 37.432 301.94 0.000
Linear 4 329.024 82.256 663.50 0.000
A 1 69.921 69.921 564.01 0.000
B 1 130.468 130.468 1052.40 0.000
C 1 99.899 99.899 805.82 0.000
D 1 28.736 28.736 231.79 0.000
2-Way Interactions 6 156.521 26.087 210.42 0.000
A*B 1 31.892 31.892 257.25 0.000
A*C 1 124.520 124.520 1004.42 0.000
A*D 1 0.002 0.002 0.01 0.912
B*C 1 0.009 0.009 0.07 0.791
B*D 1 0.096 0.096 0.78 0.391
C*D 1 0.002 0.002 0.01 0.910
3-Way Interactions 4 75.910 18.978 153.08 0.000
A*B*C 1 75.645 75.645 610.18 0.000
A*B*D 1 0.206 0.206 1.66 0.216
A*C*D 1 0.053 0.053 0.43 0.521
B*C*D 1 0.006 0.006 0.05 0.832
4-Way Interactions 1 0.019 0.019 0.15 0.703
A*B*C*D 1 0.019 0.019 0.15 0.703
Error 16 1.984 0.124
Total 31 563.457
Interaction effect A*D,B*C,B*D,C*D,A*B*D,A*C*D,B*C*D and A*B*C*D >0.05 hence it is not significant. We say that this factor affects cracking.
C.
Regression Equation
Crack leng = 11.9568 - 1.4782 A_-1 + 1.4782 A_1 - 2.0192 B_-1 + 2.0192 B_1 + 1.7669 C_-1
- 1.7669 C_1 - 0.9476 D_-1 + 0.9476 D_1 + 0.9983 A*B_-1 -1 - 0.9983 A*B_-1 1
- 0.9983 A*B_1 -1 + 0.9983 A*B_1 1 - 1.9726 A*C_-1 -1 + 1.9726 A*C_-1 1
+ 1.9726 A*C_1 -1 - 1.9726 A*C_1 1 + 0.0070 A*D_-1 -1 - 0.0070 A*D_-1 1
- 0.0070 A*D_1 -1 + 0.0070 A*D_1 1 + 0.0167 B*C_-1 -1 - 0.0167 B*C_-1 1
- 0.0167 B*C_1 -1 + 0.0167 B*C_1 1 + 0.0549 B*D_-1 -1 - 0.0549 B*D_-1 1
- 0.0549 B*D_1 -1 + 0.0549 B*D_1 1 - 0.0072 C*D_-1 -1 + 0.0072 C*D_-1 1
+ 0.0072 C*D_1 -1 - 0.0072 C*D_1 1 - 1.5375 A*B*C_-1 -1 -1 + 1.5375 A*B*C_-1 -1
1 + 1.5375 A*B*C_-1 1 -1 - 1.5375 A*B*C_-1 1 1 + 1.5375 A*B*C_1 -1 -1
- 1.5375 A*B*C_1 -1 1 - 1.5375 A*B*C_1 1 -1 + 1.5375 A*B*C_1 1 1
- 0.0802 A*B*D_-1 -1 -1 + 0.0802 A*B*D_-1 -1 1 + 0.0802 A*B*D_-1 1 -1
- 0.0802 A*B*D_-1 1 1 + 0.0802 A*B*D_1 -1 -1 - 0.0802 A*B*D_1 -1 1
- 0.0802 A*B*D_1 1 -1 + 0.0802 A*B*D_1 1 1 - 0.0408 A*C*D_-1 -1 -1
+ 0.0408 A*C*D_-1 -1 1 + 0.0408 A*C*D_-1 1 -1 - 0.0408 A*C*D_-1 1 1
+ 0.0408 A*C*D_1 -1 -1 - 0.0408 A*C*D_1 -1 1 - 0.0408 A*C*D_1 1 -1
+ 0.0408 A*C*D_1 1 1 + 0.0134 B*C*D_-1 -1 -1 - 0.0134 B*C*D_-1 -1 1
- 0.0134 B*C*D_-1 1 -1 + 0.0134 B*C*D_-1 1 1 - 0.0134 B*C*D_1 -1 -1
+ 0.0134 B*C*D_1 -1 1 + 0.0134 B*C*D_1 1 -1 - 0.0134 B*C*D_1 1 1
- 0.0242 A*B*C*D_-1 -1 -1 -1 + 0.0242 A*B*C*D_-1 -1 -1 1 + 0.0242 A*B*C*D_-1 -1
1 -1 - 0.0242 A*B*C*D_-1 -1 1 1 + 0.0242 A*B*C*D_-1 1 -1 -1 - 0.0242 A*B*C*D_-1
1 -1 1 - 0.0242 A*B*C*D_-1 1 1 -1 + 0.0242 A*B*C*D_-1 1 1 1 + 0.0242 A*B*C*D_1
-1 -1 -1 - 0.0242 A*B*C*D_1 -1 -1 1 - 0.0242 A*B*C*D_1 -1 1 -1
+ 0.0242 A*B*C*D_1 -1 1 1 - 0.0242 A*B*C*D_1 1 -1 -1 + 0.0242 A*B*C*D_1 1 -1 1
+ 0.0242 A*B*C*D_1 1 1 -1 - 0.0242 A*B*C*D_1 1 1 1
D.
Residual Plots for Crack leng
From above 1st plot is normal probability plot all the points along with line hence it is normal and also from histogram we see that it look likes bell shape but not perfect bell shape but still hold it is normal and from fitted value plot we see that there is some perfect pattern hence assumption of constant variance is not hold.
Model Summary
S R-sq R-sq(adj) R-sq(pred)
0.352097 99.65% 99.32% 98.59%
from R sq value we say that model is good. it explain 99% variation hence model is good
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