The depth of wetting of a sol is the depth to which water content will increase
ID: 3222794 • Letter: T
Question
The depth of wetting of a sol is the depth to which water content will increase owing to external factors. The article "Discussion of Method for Evaluation of Depth of Wetting in Residential Areas" (J. Nelson, K. Chao, and D. Overton, Journal of Geotechnical and Geoenvironmental Engineering, 2011:293-296) discusses the relationship between depth of wetting beneath a structure and the age of the structure. The article presents measurements of depth of wetting, in meters, and the ages, in years, of 21 houses, as shown in the following table. a. Compute the least-squares line for predicting depth of wetting (y) from age (x). b. Identify a point with an unusually large x-value. Compute the least-squares line that results from deletion of this point. c. Identify another point which can be classified as an outlier. Compute the least-squares line that results from deletion of the outlier, replacing the point with the unusually large x-value. d. Which of these two points is more influential? Explain.Explanation / Answer
Answer:
a).
The regression line y=2.9122+0.8773x
Regression Analysis
r²
0.501
n
21
r
0.708
k
1
Std. Error
2.460
Dep. Var.
y
ANOVA table
Source
SS
df
MS
F
p-value
Regression
115.4571
1
115.4571
19.08
.0003
Residual
114.9610
19
6.0506
Total
230.4181
20
Regression output
confidence interval
variables
coefficients
std. error
t (df=19)
p-value
95% lower
95% upper
Intercept
2.9122
1.3192
2.208
.0398
0.1511
5.6733
x
0.8773
0.2008
4.368
.0003
0.4570
1.2977
Studentized
Studentized
Deleted
Observation
y
Predicted
Residual
Leverage
Residual
Residual
1
7.60
5.54
2.06
0.108
0.885
0.879
2
4.60
6.42
-1.82
0.074
-0.770
-0.761
3
6.10
8.18
-2.08
0.048
-0.865
-0.859
4
9.10
8.18
0.92
0.048
0.385
0.376
5
4.30
5.54
-1.24
0.108
-0.535
-0.525
6
7.30
9.93
-2.63
0.074
-1.112
-1.119
7
5.20
7.30
-2.10
0.054
-0.877
-0.872
8
10.40
9.93
0.47
0.074
0.198
0.193
9
15.50
8.18
7.32
0.048
3.051
4.158
10
5.80
4.67
1.13
0.154
0.501
0.491
11
10.70
8.18
2.52
0.048
1.051
1.054
12
5.50
6.42
-0.92
0.074
-0.389
-0.381
13
6.10
5.54
0.56
0.108
0.239
0.233
14
10.70
9.93
0.77
0.074
0.325
0.317
15
10.40
8.18
2.22
0.048
0.926
0.923
16
4.60
6.42
-1.82
0.074
-0.770
-0.761
17
7.00
9.05
-2.05
0.054
-0.858
-0.852
18
6.10
8.18
-2.08
0.048
-0.865
-0.859
19
16.80
15.19
1.61
0.474
0.900
0.895
20
9.10
11.69
-2.59
0.154
-1.143
-1.153
21
8.80
9.05
-0.25
0.054
-0.106
-0.103
b).
The largest x value is 14. After removing this point, the regression line is
Y=3.7438+0.7145x
Regression Analysis
r²
0.277
n
20
r
0.527
k
1
Std. Error
2.473
Dep. Var.
y
ANOVA table
Source
SS
df
MS
F
p-value
Regression
42.2694
1
42.2694
6.91
.0170
Residual
110.0601
18
6.1145
Total
152.3295
19
Regression output
confidence interval
variables
coefficients
std. error
t (df=18)
p-value
95% lower
95% upper
Intercept
3.7438
1.6191
2.312
.0328
0.3422
7.1455
x
0.7145
0.2717
2.629
.0170
0.1436
1.2854
c).
The point(6,15.5) is considered as outlier.
After removing this point, the regression line is
Y=2.546+0.8773x
Regression Analysis
r²
0.663
n
20
r
0.814
k
1
Std. Error
1.805
Dep. Var.
y
ANOVA table
Source
SS
df
MS
F
p-value
Regression
115.4571
1
115.4571
35.44
1.24E-05
Residual
58.6409
18
3.2578
Total
174.0980
19
Regression output
confidence interval
variables
coefficients
std. error
t (df=18)
p-value
95% lower
95% upper
Intercept
2.5460
0.9720
2.619
.0174
0.5039
4.5881
x
0.8773
0.1474
5.953
1.24E-05
0.5677
1.1870
d).
The model with removing the outlier (6,15.5) is better model. This model has larger R square than the first model.
Regression Analysis
r²
0.501
n
21
r
0.708
k
1
Std. Error
2.460
Dep. Var.
y
ANOVA table
Source
SS
df
MS
F
p-value
Regression
115.4571
1
115.4571
19.08
.0003
Residual
114.9610
19
6.0506
Total
230.4181
20
Regression output
confidence interval
variables
coefficients
std. error
t (df=19)
p-value
95% lower
95% upper
Intercept
2.9122
1.3192
2.208
.0398
0.1511
5.6733
x
0.8773
0.2008
4.368
.0003
0.4570
1.2977
Studentized
Studentized
Deleted
Observation
y
Predicted
Residual
Leverage
Residual
Residual
1
7.60
5.54
2.06
0.108
0.885
0.879
2
4.60
6.42
-1.82
0.074
-0.770
-0.761
3
6.10
8.18
-2.08
0.048
-0.865
-0.859
4
9.10
8.18
0.92
0.048
0.385
0.376
5
4.30
5.54
-1.24
0.108
-0.535
-0.525
6
7.30
9.93
-2.63
0.074
-1.112
-1.119
7
5.20
7.30
-2.10
0.054
-0.877
-0.872
8
10.40
9.93
0.47
0.074
0.198
0.193
9
15.50
8.18
7.32
0.048
3.051
4.158
10
5.80
4.67
1.13
0.154
0.501
0.491
11
10.70
8.18
2.52
0.048
1.051
1.054
12
5.50
6.42
-0.92
0.074
-0.389
-0.381
13
6.10
5.54
0.56
0.108
0.239
0.233
14
10.70
9.93
0.77
0.074
0.325
0.317
15
10.40
8.18
2.22
0.048
0.926
0.923
16
4.60
6.42
-1.82
0.074
-0.770
-0.761
17
7.00
9.05
-2.05
0.054
-0.858
-0.852
18
6.10
8.18
-2.08
0.048
-0.865
-0.859
19
16.80
15.19
1.61
0.474
0.900
0.895
20
9.10
11.69
-2.59
0.154
-1.143
-1.153
21
8.80
9.05
-0.25
0.054
-0.106
-0.103