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 1 year ago

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