Problem # 3, exercise 3.8, page 122 (table B.5, Belle Ayr Liquefaction Runs) Ple
ID: 3043404 • Letter: P
Question
Problem # 3, exercise 3.8, page 122 (table B.5, Belle Ayr Liquefaction Runs) Please use R, All codes must be shown.
The data in Table B.5 present the performance of a chemical process as a function of sever controllable process variables.
a.- Fit a multiple regression model relating CO2 product (y) to total solvent (x6) and hydrogen consumption (x7).
b.- Test for significance of regression. Calculate R^2 and R^2 Adj.
c.- Using t tests determine the contribution of x6 and x7 to the model.
d.- Construct 95% CIs on B6 and B7.
e.- Refit the model using only x6 as the regressor. Test for significance of regression and calculate R^2 and R^2 Adj. Discuss your findings. Based on these statistics, are you satisfied with this model?
f.- Construct a 95% CI on B6 using the model you fit in part e. Compare the length of this CI to the length of the CI in part d. Does this tell you anything important about the contribution of x7 to the model?
g.- Compare the values of MSres obtained for the two models you have fit (parts a and e). How did the MSres change when you removed x7 from the model? Does this tell you anything important about the contribution of x7 to the model?
Table B.5. (Please use R, and show the codes).
y
x1
x2
x3
x4
x5
x6
x7
36.98
5.1
400
51.37
4.24
1484.83
2227.25
2.06
13.74
26.4
400
72.33
30.87
289.94
434.9
1.33
10.08
23.8
400
71.44
33.01
320.79
481.19
0.97
8.53
46.4
400
79.15
44.61
164.76
247.14
0.62
36.42
7
450
80.47
33.84
1097.26
1645.89
0.22
26.59
12.6
450
89.9
41.26
605.06
907.59
0.76
19.07
18.9
450
91.48
41.88
405.37
608.05
1.71
5.96
30.2
450
98.6
70.79
253.7
380.55
3.93
15.52
53.8
450
98.05
66.82
142.27
213.4
1.97
56.61
5.6
400
55.69
8.92
1362.24
2043.36
5.08
26.72
15.1
400
66.29
17.98
507.65
761.48
0.6
20.8
20.3
400
58.94
17.79
377.6
566.4
0.9
6.99
48.4
400
74.74
33.94
158.05
237.08
0.63
45.93
5.8
425
63.71
11.95
130.66
1961.49
2.04
43.09
11.2
425
67.14
14.73
682.59
1023.89
1.57
15.79
27.9
425
77.65
34.49
274.2
411.3
2.38
21.6
5.1
450
67.22
14.48
1496.51
2244.77
0.32
35.19
11.7
450
81.48
29.69
652.43
978.64
0.44
26.14
16.7
450
83.88
26.33
458.42
687.62
8.82
8.6
24.8
450
89.38
37.98
312.25
468.38
0.02
11.63
24.9
450
79.77
25.66
307.08
460.62
1.72
9.59
39.5
450
87.93
22.36
193.61
290.42
1.88
4.42
29
450
79.5
31.52
155.96
233.95
1.43
38.89
5.5
460
72.73
17.86
1392.08
2088.12
1.35
11.19
11.5
450
77.88
25.2
663.09
994.63
1.61
75.62
5.2
470
75.5
8.66
1464.11
2196.17
4.78
36.03
10.6
470
83.15
22.39
720.07
1080.11
5.88
y: CO2
x1: Space time, min.
x2: Temperature, oC
x3: Percent solvation
x4:Oil yield (g/100 g MAF)
x5: Coal total
x6: Solvent total
x7: Hydrogen consumption
y
x1
x2
x3
x4
x5
x6
x7
36.98
5.1
400
51.37
4.24
1484.83
2227.25
2.06
13.74
26.4
400
72.33
30.87
289.94
434.9
1.33
10.08
23.8
400
71.44
33.01
320.79
481.19
0.97
8.53
46.4
400
79.15
44.61
164.76
247.14
0.62
36.42
7
450
80.47
33.84
1097.26
1645.89
0.22
26.59
12.6
450
89.9
41.26
605.06
907.59
0.76
19.07
18.9
450
91.48
41.88
405.37
608.05
1.71
5.96
30.2
450
98.6
70.79
253.7
380.55
3.93
15.52
53.8
450
98.05
66.82
142.27
213.4
1.97
56.61
5.6
400
55.69
8.92
1362.24
2043.36
5.08
26.72
15.1
400
66.29
17.98
507.65
761.48
0.6
20.8
20.3
400
58.94
17.79
377.6
566.4
0.9
6.99
48.4
400
74.74
33.94
158.05
237.08
0.63
45.93
5.8
425
63.71
11.95
130.66
1961.49
2.04
43.09
11.2
425
67.14
14.73
682.59
1023.89
1.57
15.79
27.9
425
77.65
34.49
274.2
411.3
2.38
21.6
5.1
450
67.22
14.48
1496.51
2244.77
0.32
35.19
11.7
450
81.48
29.69
652.43
978.64
0.44
26.14
16.7
450
83.88
26.33
458.42
687.62
8.82
8.6
24.8
450
89.38
37.98
312.25
468.38
0.02
11.63
24.9
450
79.77
25.66
307.08
460.62
1.72
9.59
39.5
450
87.93
22.36
193.61
290.42
1.88
4.42
29
450
79.5
31.52
155.96
233.95
1.43
38.89
5.5
460
72.73
17.86
1392.08
2088.12
1.35
11.19
11.5
450
77.88
25.2
663.09
994.63
1.61
75.62
5.2
470
75.5
8.66
1464.11
2196.17
4.78
36.03
10.6
470
83.15
22.39
720.07
1080.11
5.88
Explanation / Answer
Output after fitting regression using X6 and X7.
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change
F Change
df1
df2
Sig. F Change
1
.836a
.700
.675
9.92437
.700
27.953
2
24
.000
a. Predictors: (Constant), Hydrogen Consumption, Solvent Total
ANOVAb
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
5506.277
2
2753.138
27.953
.000a
Residual
2363.835
24
98.493
Total
7870.112
26
a. Predictors: (Constant), Hydrogen Consumption, Solvent Total
b. Dependent Variable: CO2
Coefficients
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
95.0% Confidence Interval for B
B
Std. Error
Beta
Lower Bound
Upper Bound
1
(Constant)
2.526
3.610
.700
.491
-4.924
9.977
Solvent Total
.019
.003
.762
6.742
.000
.013
.024
Hydrogen Consumption
2.186
.973
.254
2.247
.034
.178
4.193
a. Dependent Variable: CO2
From the table of coefficients one can see that P value of Solvent Total = 0.000 < 0.05 and that of Hydrogen Consumption is 0.034 < 0.05 showing individual variables are also significant. Further 1 Unit of solvent Total cause change in CO2 by 0.019 units whereas 1 unit of Hydrogen Consumption causes change in CO2 by 2.186 units.
95 % CI for Hydrogen Consumption is (0.178, 4.193)
Output after fitting regression using only X6.
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change
F Change
df1
df2
Sig. F Change
1
.798a
.636
.622
10.69799
.636
43.766
1
25
.000
a. Predictors: (Constant), Solvent Total
ANOVAb
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
5008.936
1
5008.936
43.766
.000a
Residual
2861.176
25
114.447
Total
7870.112
26
a. Predictors: (Constant), Solvent Total
b. Dependent Variable: CO2
Coefficients
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
95.0% Confidence Interval for B
B
Std. Error
Beta
Lower Bound
Upper Bound
1
(Constant)
6.144
3.483
1.764
.090
-1.029
13.318
Solvent Total
.019
.003
.798
6.616
.000
.013
.025
a. Dependent Variable: CO2
Thus their ratio = 1.161981, it means that MS residual increases by 16.1981 % when model with single variable is used as compare to that model based on two variable.
Output after fitting regression using X6 and X7.
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change
F Change
df1
df2
Sig. F Change
1
.836a
.700
.675
9.92437
.700
27.953
2
24
.000
a. Predictors: (Constant), Hydrogen Consumption, Solvent Total
ANOVAb
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
5506.277
2
2753.138
27.953
.000a
Residual
2363.835
24
98.493
Total
7870.112
26
a. Predictors: (Constant), Hydrogen Consumption, Solvent Total
b. Dependent Variable: CO2
Coefficients
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
95.0% Confidence Interval for B
B
Std. Error
Beta
Lower Bound
Upper Bound
1
(Constant)
2.526
3.610
.700
.491
-4.924
9.977
Solvent Total
.019
.003
.762
6.742
.000
.013
.024
Hydrogen Consumption
2.186
.973
.254
2.247
.034
.178
4.193
a. Dependent Variable: CO2
- From ANOVA table one can read the P-value =0.000 <0.05 indicates that regression is significant, means Solvent Total and Hydrogen Consumption can be used to predict CO2.
- Contribution of Hydrogen Consumption, Solvent Total.
From the table of coefficients one can see that P value of Solvent Total = 0.000 < 0.05 and that of Hydrogen Consumption is 0.034 < 0.05 showing individual variables are also significant. Further 1 Unit of solvent Total cause change in CO2 by 0.019 units whereas 1 unit of Hydrogen Consumption causes change in CO2 by 2.186 units.
- 95 % CI for Solvent Total is (0.013, 0.024) with length = 0.024 – 0.013 = 0.011
95 % CI for Hydrogen Consumption is (0.178, 4.193)
- R2 =0.700 and Adjusted R2 = 0.675
Output after fitting regression using only X6.
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change
F Change
df1
df2
Sig. F Change
1
.798a
.636
.622
10.69799
.636
43.766
1
25
.000
a. Predictors: (Constant), Solvent Total
ANOVAb
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
5008.936
1
5008.936
43.766
.000a
Residual
2861.176
25
114.447
Total
7870.112
26
a. Predictors: (Constant), Solvent Total
b. Dependent Variable: CO2
Coefficients
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
95.0% Confidence Interval for B
B
Std. Error
Beta
Lower Bound
Upper Bound
1
(Constant)
6.144
3.483
1.764
.090
-1.029
13.318
Solvent Total
.019
.003
.798
6.616
.000
.013
.025
a. Dependent Variable: CO2
- From table of ANOVA one can see that P-value =0.000 < 0.05 showing that regression is significant.
- R2 = 0.636 Adjusted R2 = 0.622
- On comparing regression with Solvent Total and Hydrogen Consumption with that with Solvent Total only we conclude that effect of Solvent Total in both model is same. There is no much difference in either R2 and Adjusted R2 , it means that model with single variable is also sufficient.
- 95 % CI for Solvent Total is (0.013, 0.025) with length = 0.025 – 0.013 = 0.012
- From point 3 and 8 it is seen that length of CI for Solvent Total is almost same.
- MS Residual for two variable model = 98.493 and MS Residual for one variable model = 114.447.
Thus their ratio = 1.161981, it means that MS residual increases by 16.1981 % when model with single variable is used as compare to that model based on two variable.
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