A business statistics professor at a college would like to develop a regression
ID: 3358685 • Letter: A
Question
A business statistics professor at a college would like to develop a regression model to predict the final scores for students based on their current GPAs, the number of hours they studied for pracitce, and the number of times they were absent during the semester. The data for these variables are in the accompanying table. Complete parts a through d below.
a. Construct a regression model using all three independent variables. Let y be the final exam scores,x1 be the GPAs,x2be the number of hours spent studying, and x3 be the number of absences during the semester.
Score
GPA
Hours
Absences
69
2.52
3.0
0
70
2.26
4.0
3
70
2.60
2.5
1
71
3.09
0.5
0
74
3.08
6.0
4
77
2.76
3.5
7
77
3.35
1.5
0
77
2.99
3.0
3
77
2.98
2.0
3
80
2.85
2.5
2
78
2.80
4.5
0
81
3.44
7.0
1
82
3.25
3.0
1
84
3.15
3.0
4
84
3.16
5.5
0
83
2.95
2.0
0
85
2.70
4.0
1
86
3.20
4.5
3
86
3.76
2.0
0
85
3.55
3.5
2
86
2.93
6.0
1
85
3.03
6.5
1
86
3.16
5.0
3
86
3.89
7.5
4
88
3.53
4.0
0
88
3.31
6.5
1
90
3.67
5.0
0
89
2.88
3.5
1
91
3.39
6.0
1
91
3.21
4.5
2
91
3.80
7.0
0
91
3.93
6.0
2
92
3.99
5.0
0
91
3.58
6.5
1
92
2.98
4.0
2
94
3.27
6.5
0
97
2.88
3.5
0
98
3.73
5.0
1
100
3.48
6.5
1
101
3.02
7.0
0
Score
GPA
Hours
Absences
ModifyingAbove y =( ) + ( ) x1 +( ) x2 + ( )x3
(Round to three decimal places as needed.)
b. Calculate the multiple coefficient of determination.
c. Test the significance of the overall regression model using =0.05
d. Calculate the adjusted multiple coefficient of determination
Score
GPA
Hours
Absences
69
2.52
3.0
0
70
2.26
4.0
3
70
2.60
2.5
1
71
3.09
0.5
0
74
3.08
6.0
4
77
2.76
3.5
7
77
3.35
1.5
0
77
2.99
3.0
3
77
2.98
2.0
3
80
2.85
2.5
2
78
2.80
4.5
0
81
3.44
7.0
1
82
3.25
3.0
1
84
3.15
3.0
4
84
3.16
5.5
0
83
2.95
2.0
0
85
2.70
4.0
1
86
3.20
4.5
3
86
3.76
2.0
0
85
3.55
3.5
2
86
2.93
6.0
1
85
3.03
6.5
1
86
3.16
5.0
3
86
3.89
7.5
4
88
3.53
4.0
0
88
3.31
6.5
1
90
3.67
5.0
0
89
2.88
3.5
1
91
3.39
6.0
1
91
3.21
4.5
2
91
3.80
7.0
0
91
3.93
6.0
2
92
3.99
5.0
0
91
3.58
6.5
1
92
2.98
4.0
2
94
3.27
6.5
0
97
2.88
3.5
0
98
3.73
5.0
1
100
3.48
6.5
1
101
3.02
7.0
0
Score
GPA
Hours
Absences
Explanation / Answer
Solution:
Required multiple regression model (by using excel) is given as below:
Regression Statistics
Multiple R
0.6833
R Square
0.4669
Adjusted R Square
0.4225
Standard Error
6.2059
Observations
40
ANOVA
df
SS
MS
F
P-value
Regression
3
1214.309
404.770
10.510
0.000
Residual
36
1386.466
38.513
Total
39
2600.775
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
55.802
8.289
6.732
0.000
38.992
72.613
GPA
7.047
2.717
2.594
0.014
1.537
12.556
Hours
1.842
0.606
3.038
0.004
0.612
3.072
Absences
-1.099
0.647
-1.699
0.098
-2.410
0.213
Part a
Required regression equation is given as below:
Y = 55.802 + 7.047*X1 + 1.842*X2 – 1.099*X3
Score = 55.802 + 7.047*GPA + 1.842*Hours – 1.099*Absences
Part b
Multiple coefficient of determination is given as below:
Coefficient of determination = R2 = R*R = 0.6833*0.6833 = 0.466899
Part c
We are given
Level of significance = = 0.05
P-value for overall regression model is given as below:
P-value = 0.00
P-value < = 0.05
So, we reject the null hypothesis that the given regression model is not statistically significant.
This means, there is sufficient evidence to conclude that given regression model is statistically significant. There is a significant linear relationship exists between the dependent variable score and set of independent variables such as GPA, Hours, and Absences.
Part d
The formula for adjusted multiple coefficient of determination is given as below:
R2adjusted = 1 – ( 1 – R2)*[(n – 1)/(n – (k + 1))]
Where, n is sample size and k is number of independent variables in regression model.
We are given
R2 = 0.466899
n = 40
k = 3
R2adjusted = 1 – ( 1 – 0.466899)*[(40 – 1)/(40 – (3 + 1))]
R2adjusted = 1 – 0.533101*[39/36]
R2adjusted = 1 – 0.533101* 1.083333
R2adjusted = 1 – 0.577526
R2adjusted = 0.422474
Regression Statistics
Multiple R
0.6833
R Square
0.4669
Adjusted R Square
0.4225
Standard Error
6.2059
Observations
40
ANOVA
df
SS
MS
F
P-value
Regression
3
1214.309
404.770
10.510
0.000
Residual
36
1386.466
38.513
Total
39
2600.775
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
55.802
8.289
6.732
0.000
38.992
72.613
GPA
7.047
2.717
2.594
0.014
1.537
12.556
Hours
1.842
0.606
3.038
0.004
0.612
3.072
Absences
-1.099
0.647
-1.699
0.098
-2.410
0.213
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