Stat 401
Lab #8 Problem1 e) and f)
Regression of Hardness on Elapsed Time
Residual vs x
26
6
25
4
2
Residual
Hardness
24
23
0
-2
22
-4
21
-6
15
20
19
15
25
20
30
35
20
25
30
35
40
40
Elapsed Time
Elapsed Time
Residual by Predicted Plot
Linear Fit
Hardness = 168.8 + 2.025*Elapsed Time
6
4
Hardness Residual
The two residual plots do slow a slight curvature pattern,
Indicating that the model may be inadequate. However, the
pattern is more likely an artifact of the actual placement of the
residuals rather than the residuals being a function of the x-variable,
That is the rsiduals do not show a consistent pattern of increase/
Decrease as x increases or decreases. The result of the lack of fit
test confirm this conclusion.
2
0
-2
-4
-6
190
210
200
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.971281
0.969229
3.329522
225.5
16
Lack Of Fit
Source
Lack Of Fit
Pure Error
Total Error
DF
2
12
14
Sum of Squares
19.20000
136.00000
155.20000
Mean Square
9.6000
11.3333
F Ratio
0.8471
Prob > F
0.4528
Max RSq
0.9748
Analysis of Variance
Source
Model
Error
C. Total
220
230
Hardness Predicted
DF
1
14
15
Sum of Squares
5248.8000
155.2000
5404.0000
Mean Square
5248.80
11.09
F Ratio
473.4742
Prob > F
<.0001*
240
250
2
Parameter Estimates
Term
Intercept
Elapsed Time
Estimate
168.8
2.025
Hardness
Elapsed Time
199
205
196
200
218
220
215
223
237
234
235
230
250
248
253
245
16
16
16
16
24
24
24
24
32
32
32
32
40
40
40
40
Std Error
2.735481
0.093063
Residuals
Hardness
-2.2
3.8
-5.2
-1.2
0.6
2.6
-2.4
5.6
3.4
0.4
1.4
-3.6
0.2
-1.8
3.2
-4.8
t Ratio
61.71
21.76
Prob>|t|
<.0001*
<.0001*
Predicted
Hardness
201.2
201.2
201.2
201.2
217.4
217.4
217.4
217.4
233.6
233.6
233.6
233.6
249.8
249.8
249.8
249.8
Problem#2
Full Model (y,x1,x2,x3)
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Analysis of Variance
Source
DF
Model
Error
C. Total
Sum of
Squares
1890.4081
178.8300
2069.2381
3
17
20
Parameter Estimates
Term
Estimate
Intercept -39.91967
x1
0.7156402
x2
1.2952861
x3
-0.152123
0.913577
0.898326
3.243364
17.52381
21
Mean Square
F Ratio
630.136
10.519
59.9022
Prob > F
<.0001
Std Error t Ratio Prob>|t| Lower 95% Upper 95%
VIF
11.896
-3.36 0.0038 -65.01803 -14.82131
.
0.134858
5.31 <.0001 0.4311143 1.0001661 2.9064836
0.368024
3.52 0.0026 0.5188228 2.0717495 2.5726324
0.156294
-0.97 0.3440 -0.481874 0.1776291 1.3335875
Residual by Predicted Plot
7.5
y Residual
5.0
2.5
0.0
-2.5
-5.0
-7.5
5
10 15 20 25 30 35 40 45
y Predicted
Normal Probability plot of Studentized Residuals
2
.01 .05.10 .25 .50 .75 .90.95 .99
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Normal Quantile Plot
Full Mode:l Residuals and Case Diagnostic Statistics
x1
80
80
75
62
62
62
62
62
58
58
58
58
58
58
50
50
50
50
50
56
70
65
x2
27
27
25
24
22
23
24
24
23
18
18
17
18
19
18
18
19
19
20
20
20
21
x3
89
88
90
87
87
87
93
93
87
80
89
88
82
93
89
86
72
79
80
82
91
87
y
42
37
37
28
18
18
19
20
15
14
14
13
11
12
8
7
8
8
9
15
15
.
Predicted y
38.77
38.92
32.44
22.30
19.71
21.01
21.39
21.39
18.14
12.73
11.36
10.22
12.43
12.05
5.64
6.09
9.52
8.46
9.60
13.59
22.24
20.56
Residual y
3.23464
-1.91749
4.55553
5.69777
-1.71165
-3.00694
-2.38949
-1.38949
-3.14438
1.26719
2.63630
2.77946
-1.42856
-0.05050
2.36142
0.90505
-1.51995
-0.45509
-0.59826
1.41215
-7.23771
.
Studentized Lower 95% Upper 95% Lower 95% Upper 95%
Resid
Mean y
Mean y
Indiv y
Indiv y
1.19334
35.01
42.52
30.96
46.57
-0.71580
35.06
42.78
31.06
46.77
1.54602
29.59
35.30
25.03
39.86
1.88182
19.85
24.76
15.03
29.57
-0.54208
18.15
21.28
12.69
26.73
-0.96526
19.10
22.91
13.90
28.11
-0.83378
18.19
24.59
13.83
28.95
-0.48484
18.19
24.59
13.83
28.95
-1.04553
15.58
20.71
10.84
25.45
0.43683
9.67
15.79
5.24
20.23
0.88426
8.67
14.06
4.01
18.72
0.96857
7.03
13.41
2.67
17.77
-0.47987
9.71
15.14
5.07
19.79
-0.01747
8.95
15.15
4.54
19.56
0.80921
2.65
8.62
-1.83
13.10
0.29935
3.62
8.57
-1.18
13.37
-0.61121
5.13
13.91
1.39
17.65
-0.15315
5.71
11.20
1.08
15.83
-0.20302
6.74
12.46
2.18
17.01
0.45398
11.65
15.53
6.48
20.70
-2.63822
18.59
25.89
14.48
29.99
.
18.58
22.54
13.44
27.69
Model (y,x1)
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Analysis of Variance
Source
DF
Model
Error
C. Total
1
19
20
0.845781
0.837664
4.098242
17.52381
21
Sum of
Squares
1750.1220
319.1161
2069.2381
Parameter Estimates
Term
Estimate
Intercept
-44.13202
x1
1.0203093
Mean Square
F Ratio
1750.12
16.80
104.2013
Prob > F
<.0001
Std Error
6.105858
0.099953
t Ratio
-7.23
10.21
Prob>|t|
<.0001
<.0001
Model (y,x1,x2)
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Analysis of Variance
Source
DF
Model
Error
C. Total
2
18
20
0.908761
0.898623
3.238615
17.52381
21
Sum of
Squares
1880.4428
188.7953
2069.2381
Parameter Estimates
Term
Estimate
Intercept
-50.35884
x1
0.6711544
x2
1.2953514
Std Error
5.138328
0.126691
0.367485
Mean Square
F Ratio
940.221
10.489
89.6420
Prob > F
<.0001
t Ratio
-9.80
5.30
3.52
Prob>|t|
<.0001
<.0001
0.0024