For the data below, the regression model ModifyingAbove Upper Y with caret Subsc
ID: 3357805 • Letter: F
Question
For the data below, the regression model
ModifyingAbove Upper Y with caret Subscript iYiequals=51.6903651.69036negative 0.014250.01425Upper X Subscript 1 iX1inegative 0.007510.00751Upper X Subscript 2 iX2i
predicts gas mileage (in miles per gallon) based on the horsepower of a car,
Upper X 1 commaX1,
and the car's weight,
Upper X 2 commaX2,
(in pounds). Develop a regression model that includes horsepower, weight, and their interaction to predict gas mileage. Complete parts (a) and (b).
3871At the
0.050.05
level of significance, is there evidence that the interaction term makes a significant contribution to the model?
Which regression model is more appropriate, the original given model or the new model with the interaction term? Explain.
MPG Horsepower Weight 15.6 190 4723 19.4 101 3530 20.4 140 3226 18.5 173 4472 17.5 167 4293 27.3 75 3187 44.4 68 2106 27.6 85 2495 28.4 91 2610 21.3 1363871At the
0.050.05
level of significance, is there evidence that the interaction term makes a significant contribution to the model?
Which regression model is more appropriate, the original given model or the new model with the interaction term? Explain.
Explanation / Answer
Solution:
First of all we have to include the interaction term X1X2 in the given data. Data with interaction is given as below:
MPG (Y)
Horsepower (X1)
Weight (X2)
Interaction (X1X2)
15.6
190
4723
897370
19.4
101
3530
356530
20.4
140
3226
451640
18.5
173
4472
773656
17.5
167
4293
716931
27.3
75
3187
239025
44.4
68
2106
143208
27.6
85
2495
212075
28.4
91
2610
237510
21.3
136
3871
526456
Now, we have to develop the new regression model with the interaction term. The new modified regression model with interaction term is given as below:
Modified model with interaction:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.94191983
R Square
0.88721297
Adjusted R Square
0.83081945
Standard Error
3.47886472
Observations
10
ANOVA
df
SS
MS
F
Significance F
Regression
3
571.2090018
190.403
15.732535
0.003002636
Residual
6
72.61499824
12.1025
Total
9
643.824
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
96.3792975
16.49052095
5.844527
0.0011065
56.02844648
136.7301486
Horsepower (X1)
-0.47017908
0.172181948
-2.730711
0.0341538
-0.891493125
-0.048865026
Weight (X2)
-0.01935799
0.005237462
-3.696062
0.0101356
-0.032173595
-0.006542377
Interaction (X1X2)
0.00011443
4.01275E-05
2.8516
0.0291205
1.62391E-05
0.000212616
Original Model:
Regression Statistics
Multiple R
0.856945774
R Square
0.73435606
Adjusted R Square
0.658457791
Standard Error
4.942930654
Observations
10
ANOVA
df
SS
MS
F
P-value
Regression
2
472.7960558
236.398
9.675531
0.009661614
Residual
7
171.0279442
24.43256
Total
9
643.824
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
51.69036048
7.292189096
7.088456
0.000196
34.4470733
68.93364766
Horsepower (X1)
-0.014251485
0.090794832
-0.15696
0.879705
-0.22894715
0.200444175
Weight (X2)
-0.007505325
0.004527699
-1.65765
0.141355
-0.01821163
0.003200982
Questions:
Part a
At the 0.05 level ofsignificance, is there evidence that the interaction term makes a significant contribution to the model?
Answer:
The p-value for the coefficient of interaction term is given as 0.0291205 which is less than the given level of significance or alpha value 0.05, so we reject the null hypothesis that there is no sufficient evidence that the interaction term makes a significant contribution to the model. This means, there is sufficient evidence to conclude that the interaction term makes a significant contribution to the model.
Part b
Which regression model is more appropriate, the original given model or the new model with the interaction term? Explain.
Answer:
The second modified regression equation is more appropriate and statistically significant because the p-value for the modified regression model with interaction term is given as 0.003002636 which is less the p-value for the original model which is given as 0.009662.
MPG (Y)
Horsepower (X1)
Weight (X2)
Interaction (X1X2)
15.6
190
4723
897370
19.4
101
3530
356530
20.4
140
3226
451640
18.5
173
4472
773656
17.5
167
4293
716931
27.3
75
3187
239025
44.4
68
2106
143208
27.6
85
2495
212075
28.4
91
2610
237510
21.3
136
3871
526456