A nickel-titanium alloy is used to make components for jet turbine aircraft engi

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 nonrecoverable failure. A test is run at the parts producer to determine the effect of four factors on cracks. The four factors are pouring temperature (A), titaniunm content (B), heat treatment method (C), and amount of grain refiner used (D). Two replicates of a 24 design are run, and the length of crack (in mm) induced in a sample coupon subjected to a standard test is measured The data are shown below Actual Actual Crack Run Order A B c D Length Run Order A B CD Length Crack 17 21 20 26 29 1.97 2.52 1.93 1.99 1.70 1.54 1.75 1.61 + 2.49 +2.19 +2.09 1.70 + t 2.19 1.58 1.68 +t2.12 2.12 2.78 1.89 1.96 1.75 1.91 1.88 1.70 + 2.62 + 2.44 + 2.43 1.98 + + 2.53 +t 2.11 1.91 1.91 25 18 27 19 13 32 23 16 24 14 12 15 28 4 30 10

Explanation / Answer

(a)

Factorial Fit: Crack length versus A, B, C, D

Estimated Effects and Coefficients for Crack length (coded units)

Term Effect Coef SE Coef T P
Constant 2.0303 0.03162 64.21 0.000<0.05
A -0.0556 -0.0278 0.03162 -0.88 0.392>0.05
B -0.2444 -0.1222 0.03162 -3.86 0.001<0.05
C -0.3269 -0.1634 0.03162 -5.17 0.000<0.05
D 0.1856 0.0928 0.03162 2.94 0.010<0.05
A*B -0.0181 -0.0091 0.03162 -0.29 0.778>0.05
A*C -0.0581 -0.0291 0.03162 -0.92 0.372>0.05
A*D -0.1831 -0.0916 0.03162 -2.90 0.011<0.05
B*C 0.1506 0.0753 0.03162 2.38 0.030<0.05
B*D -0.0469 -0.0234 0.03162 -0.74 0.469>0.05
C*D 0.0881 0.0441 0.03162 1.39 0.183>0.05
A*B*C 0.1619 0.0809 0.03162 2.56 0.021<0.05
A*B*D 0.1569 0.0784 0.03162 2.48 0.025<0.05
A*C*D 0.1494 0.0747 0.03162 2.36 0.031<0.05
B*C*D -0.0569 -0.0284 0.03162 -0.90 0.382>0.05
A*B*C*D 0.0669 0.0334 0.03162 1.06 0.306>0.05

Main effects B, C, D and interaction effects AD, BC, ABC, ABD, ACD are significant since their corresponding p-value<0.05.

(b)

Analysis of Variance for Crack length (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 4 1.63294 1.63294 0.408234 12.76 0.000
A 1 0.02475 0.02475 0.024753 0.77 0.392
B 1 0.47775 0.47775 0.477753 14.93 0.001
C 1 0.85478 0.85478 0.854778 26.71 0.000
D 1 0.27565 0.27565 0.275653 8.62 0.010
2-Way Interactions 6 0.55914 0.55914 0.093191 2.91 0.041
A*B 1 0.00263 0.00263 0.002628 0.08 0.778
A*C 1 0.02703 0.02703 0.027028 0.84 0.372
A*D 1 0.26828 0.26828 0.268278 8.38 0.011
B*C 1 0.18150 0.18150 0.181503 5.67 0.030
B*D 1 0.01758 0.01758 0.017578 0.55 0.469
C*D 1 0.06213 0.06213 0.062128 1.94 0.183
3-Way Interactions 4 0.61089 0.61089 0.152722 4.77 0.010
A*B*C 1 0.20963 0.20963 0.209628 6.55 0.021
A*B*D 1 0.19688 0.19688 0.196878 6.15 0.025
A*C*D 1 0.17850 0.17850 0.178503 5.58 0.031
B*C*D 1 0.02588 0.02588 0.025878 0.81 0.382
4-Way Interactions 1 0.03578 0.03578 0.035778 1.12 0.306
A*B*C*D 1 0.03578 0.03578 0.035778 1.12 0.306
Residual Error 16 0.51195 0.51195 0.031997
Pure Error 16 0.51195 0.51195 0.031997
Total 31 3.35070

(c)

Unusual Observations for Crack length

Crack
Obs StdOrder length Fit SE Fit Residual St Resid
14 14 1.58000 1.84500 0.12648 -0.26500 -2.10R
30 30 2.11000 1.84500 0.12648 0.26500 2.10R

R denotes an observation with a large standardized residual.

(d)

Model:

Crack length (y)=2.0303-0.2444B-0.3269C+0.1856D-0.1831AD+0.1506BC+0.1619ABC+0.1569ABD+0.1496ACD+Error

and its R-Sq = 84.72% and R-Sq(adj) = 70.40%

hence the model is quite satisfactory.

(e)

Crack
Obs StdOrder length Fit SE Fit Residual St Resid
1 1 1.97000 2.04500 0.12648 -0.07500 -0.59
2 2 2.52000 2.65000 0.12648 -0.13000 -1.03
3 3 1.93000 1.91000 0.12648 0.02000 0.16
4 4 1.99000 1.97500 0.12648 0.01500 0.12
5 5 1.70000 1.72500 0.12648 -0.02500 -0.20
6 6 1.54000 1.72500 0.12648 -0.18500 -1.46
7 7 1.75000 1.81500 0.12648 -0.06500 -0.51
8 8 1.61000 1.65500 0.12648 -0.04500 -0.36
9 9 2.49000 2.55500 0.12648 -0.06500 -0.51
10 10 2.19000 2.31500 0.12648 -0.12500 -0.99
11 11 2.09000 2.26000 0.12648 -0.17000 -1.34
12 12 1.70000 1.84000 0.12648 -0.14000 -1.11
13 13 2.19000 2.36000 0.12648 -0.17000 -1.34
14 14 1.58000 1.84500 0.12648 -0.26500 -2.10R
15 15 1.68000 1.79500 0.12648 -0.11500 -0.91
16 16 2.12000 2.01500 0.12648 0.10500 0.83
17 17 2.12000 2.04500 0.12648 0.07500 0.59
18 18 2.78000 2.65000 0.12648 0.13000 1.03
19 19 1.89000 1.91000 0.12648 -0.02000 -0.16
20 20 1.96000 1.97500 0.12648 -0.01500 -0.12
21 21 1.75000 1.72500 0.12648 0.02500 0.20
22 22 1.91000 1.72500 0.12648 0.18500 1.46
23 23 1.88000 1.81500 0.12648 0.06500 0.51
24 24 1.70000 1.65500 0.12648 0.04500 0.36
25 25 2.62000 2.55500 0.12648 0.06500 0.51
26 26 2.44000 2.31500 0.12648 0.12500 0.99
27 27 2.43000 2.26000 0.12648 0.17000 1.34
28 28 1.98000 1.84000 0.12648 0.14000 1.11
29 29 2.53000 2.36000 0.12648 0.17000 1.34
30 30 2.11000 1.84500 0.12648 0.26500 2.10R
31 31 1.91000 1.79500 0.12648 0.11500 0.91
32 32 1.91000 2.01500 0.12648 -0.10500 -0.83

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.