Question

In: Statistics and Probability

The following data set is taken from an experiment performed at NIST. The purpose is to...

The following data set is taken from an experiment performed at NIST. The purpose is to determine the effect of machining factors on ceramic strength.

Number of observations = 32 (a full 25 factorial design)

Response Variable Y = Mean (over 15 reps) of Ceramic Strength

Factor 1 = Table Speed (2 levels: slow (.025 m/s) and fast (.125 m/s))

Factor 2 = Down Feed Rate (2 levels: slow (.05 mm) and fast (.125 mm))

Factor 3 = Wheel Grit (2 levels: 140/170 and 80/100)

Factor 4 = Direction (2 levels: longitudinal and transverse)

Factor 5 = Batch (2 levels: 1 and 2)

Run the experiment and analyze the results. Determine what 2-way interactions and main effects Plot the main effects and the significant 2-way interactions and add your conclusions.

The design matrix is given in an actual randomized run order:

(X1:Table Speed) (X2:Feed Rate) (X3:Wheel Grit) (X4:Direction)

(X5:Batch)

 (Y:Strength) (Actual Run Order)
-1 -1 -1 -1 -1 476.315 17
1 -1 -1 -1 -1 505.736 30
-1 1 -1 -1 -1 491.498 14
1 1 -1 -1 -1 466.851 8
-1 -1 1 -1 -1 492.569 32
1 -1 1 -1 -1 449.498 20
-1 1 1 -1 -1 485.086 26
1 1 1 -1 -1 468.482 24
-1 -1 -1 1 -1 344.106 10
1 -1 -1 1 -1 332.864 16
-1 1 -1 1 -1 335.132 27
1 1 -1 1 -1 397.761 18
-1 -1 1 1 -1 311.304 3
1 -1 1 1 -1 287.259 19
-1 1 1 1 -1 299.957 31
1 1 1 1 -1 344.029 15
-1 -1 -1 -1 1 425.138 12
1 -1 -1 -1 1 434.56 1
-1 1 -1 -1 1 427.385 4
1 1 -1 -1 1 446.628 23
-1 -1 1 -1 1 409.633 2
1 -1 1 -1 1 410.319 28
-1 1 1 -1 1 421.169 11
1 1 1 -1 1 425.817 9
-1 -1 -1 1 1 310.03 25
1 -1 -1 1 1 304.087 21
-1 1 -1 1 1 292.362 6
1 1 -1 1 1 357.588 7
-1 -1 1 1 1 274.477 5
1 -1 1 1 1 240.254 13
-1 1 1 1 1 269.864 22
1 1 1 1 1 312.711 29

Solutions

Expert Solution

Estimated Effects and Coefficients for (Y:Strength) (coded units)

Term Effect Coef
Constant 382.83
X1:Table Speed) 7.40 3.70
(X2:Feed Rate) 14.64 7.32
(X3:Wheel Grit) -27.85 -13.93
(X4:Direction) -138.93 -69.47
(X5:Batch) -45.40 -22.70
X1:Table Speed)*(X2:Feed Rate) 17.28 8.64
X1:Table Speed)*(X3:Wheel Grit) -10.61 -5.31
X1:Table Speed)*(X4:Direction) 10.01 5.01
X1:Table Speed)*(X5:Batch) 5.34 2.67
(X2:Feed Rate)*(X3:Wheel Grit) 4.34 2.17
(X2:Feed Rate)*(X4:Direction) 10.99 5.50
(X2:Feed Rate)*(X5:Batch) 3.49 1.75
(X3:Wheel Grit)*(X4:Direction) -13.91 -6.95
(X3:Wheel Grit)*(X5:Batch) -1.34 -0.67
(X4:Direction)*(X5:Batch) 9.02 4.51
X1:Table Speed)*(X2:Feed Rate)* 4.68 2.34
(X3:Wheel Grit)
X1:Table Speed)*(X2:Feed Rate)* 19.00 9.50
(X4:Direction)
X1:Table Speed)*(X3:Wheel Grit)* 0.36 0.18
(X4:Direction)
X1:Table Speed)*(X2:Feed Rate)* 2.98 1.49
(X5:Batch)
X1:Table Speed)*(X3:Wheel Grit)* 1.36 0.68
(X5:Batch)
X1:Table Speed)*(X4:Direction)* -5.78 -2.89
(X5:Batch)
(X2:Feed Rate)*(X3:Wheel Grit)* -1.65 -0.83
(X4:Direction)
(X2:Feed Rate)*(X3:Wheel Grit)* 1.25 0.63
(X5:Batch)
(X2:Feed Rate)*(X4:Direction)* -3.20 -1.60
(X5:Batch)
(X3:Wheel Grit)*(X4:Direction)* 1.41 0.71
(X5:Batch)
X1:Table Speed)*(X2:Feed Rate)* -4.66 -2.33
(X3:Wheel Grit)*(X4:Direction)
X1:Table Speed)*(X2:Feed Rate)* -4.67 -2.34
(X3:Wheel Grit)*(X5:Batch)
X1:Table Speed)*(X2:Feed Rate)* -2.20 -1.10
(X4:Direction)*(X5:Batch)
X1:Table Speed)*(X3:Wheel Grit)* -3.78 -1.89
(X4:Direction)*(X5:Batch)
(X2:Feed Rate)*(X3:Wheel Grit)* 4.06 2.03
(X4:Direction)*(X5:Batch)
X1:Table Speed)*(X2:Feed Rate)* 6.13 3.06
(X3:Wheel Grit)*(X4:Direction)*
(X5:Batch)

Source Seq SS
Main Effects 179263
X1:Table Speed) 438
(X2:Feed Rate) 1714
(X3:Wheel Grit) 6205
(X4:Direction) 154415
(X5:Batch) 16490
2-Way Interactions 7747
X1:Table Speed)*(X2:Feed Rate) 2388
X1:Table Speed)*(X3:Wheel Grit) 901
X1:Table Speed)*(X4:Direction) 802
X1:Table Speed)*(X5:Batch) 228
(X2:Feed Rate)*(X3:Wheel Grit) 151
(X2:Feed Rate)*(X4:Direction) 967
(X2:Feed Rate)*(X5:Batch) 98
(X3:Wheel Grit)*(X4:Direction) 1548
(X3:Wheel Grit)*(X5:Batch) 14
(X4:Direction)*(X5:Batch) 651
3-Way Interactions 3550
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit) 175
X1:Table Speed)*(X2:Feed Rate)*(X4:Direction) 2889
X1:Table Speed)*(X3:Wheel Grit)*(X4:Direction) 1
X1:Table Speed)*(X2:Feed Rate)*(X5:Batch) 71
X1:Table Speed)*(X3:Wheel Grit)*(X5:Batch) 15
X1:Table Speed)*(X4:Direction)*(X5:Batch) 267
(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction) 22
(X2:Feed Rate)*(X3:Wheel Grit)*(X5:Batch) 13
(X2:Feed Rate)*(X4:Direction)*(X5:Batch) 82
(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 16
4-Way Interactions 633
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction) 174
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X5:Batch) 175
X1:Table Speed)*(X2:Feed Rate)*(X4:Direction)*(X5:Batch) 39
X1:Table Speed)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 114
(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 132
5-Way Interactions 300
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 300
Residual Error *
Total 191492

Source Adj SS
Main Effects 179263
X1:Table Speed) 438
(X2:Feed Rate) 1714
(X3:Wheel Grit) 6205
(X4:Direction) 154415
(X5:Batch) 16490
2-Way Interactions 7747
X1:Table Speed)*(X2:Feed Rate) 2388
X1:Table Speed)*(X3:Wheel Grit) 901
X1:Table Speed)*(X4:Direction) 802
X1:Table Speed)*(X5:Batch) 228
(X2:Feed Rate)*(X3:Wheel Grit) 151
(X2:Feed Rate)*(X4:Direction) 967
(X2:Feed Rate)*(X5:Batch) 98
(X3:Wheel Grit)*(X4:Direction) 1548
(X3:Wheel Grit)*(X5:Batch) 14
(X4:Direction)*(X5:Batch) 651
3-Way Interactions 3550
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit) 175
X1:Table Speed)*(X2:Feed Rate)*(X4:Direction) 2889
X1:Table Speed)*(X3:Wheel Grit)*(X4:Direction) 1
X1:Table Speed)*(X2:Feed Rate)*(X5:Batch) 71
X1:Table Speed)*(X3:Wheel Grit)*(X5:Batch) 15
X1:Table Speed)*(X4:Direction)*(X5:Batch) 267
(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction) 22
(X2:Feed Rate)*(X3:Wheel Grit)*(X5:Batch) 13
(X2:Feed Rate)*(X4:Direction)*(X5:Batch) 82
(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 16
4-Way Interactions 633
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction) 174
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X5:Batch) 175
X1:Table Speed)*(X2:Feed Rate)*(X4:Direction)*(X5:Batch) 39
X1:Table Speed)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 114
(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 132
5-Way Interactions 300
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 300
Residual Error *
Total

Source Adj MS
Main Effects 35853
X1:Table Speed) 438
(X2:Feed Rate) 1714
(X3:Wheel Grit) 6205
(X4:Direction) 154415
(X5:Batch) 16490
2-Way Interactions 775
X1:Table Speed)*(X2:Feed Rate) 2388
X1:Table Speed)*(X3:Wheel Grit) 901
X1:Table Speed)*(X4:Direction) 802
X1:Table Speed)*(X5:Batch) 228
(X2:Feed Rate)*(X3:Wheel Grit) 151
(X2:Feed Rate)*(X4:Direction) 967
(X2:Feed Rate)*(X5:Batch) 98
(X3:Wheel Grit)*(X4:Direction) 1548
(X3:Wheel Grit)*(X5:Batch) 14
(X4:Direction)*(X5:Batch) 651
3-Way Interactions 355
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit) 175
X1:Table Speed)*(X2:Feed Rate)*(X4:Direction) 2889
X1:Table Speed)*(X3:Wheel Grit)*(X4:Direction) 1
X1:Table Speed)*(X2:Feed Rate)*(X5:Batch) 71
X1:Table Speed)*(X3:Wheel Grit)*(X5:Batch) 15
X1:Table Speed)*(X4:Direction)*(X5:Batch) 267
(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction) 22
(X2:Feed Rate)*(X3:Wheel Grit)*(X5:Batch) 13
(X2:Feed Rate)*(X4:Direction)*(X5:Batch) 82
(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 16
4-Way Interactions 127
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction) 174
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X5:Batch) 175
X1:Table Speed)*(X2:Feed Rate)*(X4:Direction)*(X5:Batch) 39
X1:Table Speed)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 114
(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 132
5-Way Interactions 300
X1:Table Speed)*(X2:Feed Rate)*(X3:Wheel Grit)*(X4:Direction)*(X5:Batch) 300
__________________________________________________________________________

Since it is 25 factorial design with single replicate, so we consider SS due 3 way , 4 way, 5 way interactions as residual hence

SSR=sum of square due to residual= 3550+633+ 300=4483 with df=10+5+1=16

MSR=SSR/16=4483/16=280.1875

_______________________________________________________________________

Observed F values

Here red line indicates the predicted Y value and black dot indicates the observed Y value.

orchestra answered 2 years ago
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