Delmarva Power is a utility company that would like to predict the monthly heati
ID: 2909801 • Letter: D
Question
Delmarva Power is a utility company that would like to predict the monthly heating bill for a household in Kent County during the month of January. A random sample of 18 households in the county were selected and their January heating bill recorded. This data is shown in the table below along with the square footage of the house (SF), the age of the heating system in years (Age,) and the type of heating system (heat pump = 1 or natural gas = 0).
Household
Bill
SF
Age
Type
1
$255
2,100
7
Natural Gas
2
$286
1,900
17
Natural Gas
3
$296
2,000
8
Natural Gas
4
$300
2,300
22
Natural Gas
5
$305
3,000
5
Natural Gas
6
$317
2,700
14
Natural Gas
7
$321
1,500
8
Natural Gas
8
$321
2,800
3
Natural Gas
9
$339
2,550
20
Natural Gas
10
$349
2,500
11
Natural Gas
11
$369
2,100
12
Heat Pump
12
$374
2,500
18
Heat Pump
13
$381
2,300
19
Heat Pump
14
$413
2,500
17
Heat Pump
15
$419
3,200
11
Heat Pump
16
$441
3,100
8
Heat Pump
17
$522
2,500
20
Heat Pump
18
$560
3,550
18
Heat Pump
Questions:
a) Develop a regression equation that will predict the monthly heating bill for a household in Kent County during the month of January based on the square footage of the house, the age of the heating system, and the type of heating system.
b) Interpret the meaning of the regression coefficients for the heating bill model.
c) Predict the monthly heating bill for a house that has 2,700 square feet and has a heat pump that is tenj years old.
d) Construct a 95% confidence interval to estimate the average monthly heating bill for a house that has 2,700 square feet and has a heat pump that is ten years old.
e) Construct a 95% prediction interval to estimate the monthly heating bill for a specific house that has 2,700 square feet and has a heat pump that is ten years old.
f) Show the calculations for the multiple coefficient of determination for the heating bill model and interpret its meaning.
g) Conduct the hypothesis test, showing the calculations, to test the significance of the overall regression model for predicting a heating bill using ? = 0.05.
h) Show the calculations for the adjusted multiple coefficient of determination for predicting a heating bill for a house in Kent County during the month of January.
i) Show the calculations for the test statistic for each regression coefficient for the heating bill model using ? = 0.05 and interpret the results.
j) Show the calculations for the 95% confidence intervals to estimate the population regression coefficients for the heating model and interpret their meaning.
Household
Bill
SF
Age
Type
1
$255
2,100
7
Natural Gas
2
$286
1,900
17
Natural Gas
3
$296
2,000
8
Natural Gas
4
$300
2,300
22
Natural Gas
5
$305
3,000
5
Natural Gas
6
$317
2,700
14
Natural Gas
7
$321
1,500
8
Natural Gas
8
$321
2,800
3
Natural Gas
9
$339
2,550
20
Natural Gas
10
$349
2,500
11
Natural Gas
11
$369
2,100
12
Heat Pump
12
$374
2,500
18
Heat Pump
13
$381
2,300
19
Heat Pump
14
$413
2,500
17
Heat Pump
15
$419
3,200
11
Heat Pump
16
$441
3,100
8
Heat Pump
17
$522
2,500
20
Heat Pump
18
$560
3,550
18
Heat Pump
Explanation / Answer
Result:
Questions:
a) Develop a regression equation that will predict the monthly heating bill for a household in Kent County during the month of January based on the square footage of the house, the age of the heating system, and the type of heating system.
The regression equation is
Bill = 145.8173+ 0.0582* SF + 2.3687* Age + 94.4710* Type
b) Interpret the meaning of the regression coefficients for the heating bill model.
When there is a one square feet increase, there is an increase of $0.0582 increase in Bill.
When there is a increase of age by 1, there is an increase of $ 2.3687 increase in Bill.
When there is a heat pump present in the house, there is an increase of $94.4710 increase in Bill.
c) Predict the monthly heating bill for a house that has 2,700 square feet and has a heat pump that is ten years old.
Predicted Bill = 145.8173+ 0.0582* 2700 + 2.3687* 10 + 94.4710* 1
=$421.053
d) Construct a 95% confidence interval to estimate the average monthly heating bill for a house that has 2,700 square feet and has a heat pump that is ten years old.
95% CI = ($379.816, $462.289)
e) Construct a 95% prediction interval to estimate the monthly heating bill for a specific house that has 2,700 square feet and has a heat pump that is ten years old.
95% PI = ($316.377, $525.729)
f) Show the calculations for the multiple coefficient of determination for the heating bill model and interpret its meaning.
R square = 83886.2096/112057.7778 = 0.7486
74.86% of variation in the Bill is explained by the model.
g) Conduct the hypothesis test, showing the calculations, to test the significance of the overall regression model for predicting a heating bill using ? = 0.05.
ANOVA table
Source
SS
df
MS
F
p-value
Regression
83,886.2096
3
27,962.0699
13.90
.0002
Residual
28,171.5681
14
2,012.2549
Total
112,057.7778
17
Calculated F= 13.90 > critical F(3,14) at 0.05 level 3.34.
Ho is rejected. The overall model is significant.
h) Show the calculations for the adjusted multiple coefficient of determination for predicting a heating bill for a house in Kent County during the month of January.
Adjusted R square = 1-(1-0.7486)*17/(18-3-1) = 0.6947
i) Show the calculations for the test statistic for each regression coefficient for the heating bill model using ? = 0.05 and interpret the results.
Test for coefficient SF, t=0.0582/0.0237 =2.456, P=0.0277 which is < 0.05 level. Ho is rejected.. SF is significant.
Test for coefficient Age, t=2.3687/2.0065 =1.180, P=0.2575 which is > 0.05 level. Ho is not rejected.. Age is not significant.
Test for coefficient Type, t=94.471/24.8928 =3.795, P=0.002 which is < 0.05 level. Ho is rejected.. Type is significant.
j) Show the calculations for the 95% confidence intervals to estimate the population regression coefficients for the heating model and interpret their meaning.
variables
coefficients
std. error
95% lower
95% upper
Intercept
145.8173
64.9862
6.4358
285.1988
SF
0.0582
0.0237
0.0074
0.1090
Age
2.3687
2.0065
-1.9348
6.6722
Type
94.4710
24.8928
41.0814
147.8607
Regression Analysis
R²
0.749
Adjusted R²
0.695
n
18
R
0.865
k
3
Std. Error
44.858
Dep. Var.
Bill
ANOVA table
Source
SS
df
MS
F
p-value
Regression
83,886.2096
3
27,962.0699
13.90
.0002
Residual
28,171.5681
14
2,012.2549
Total
112,057.7778
17
Regression output
confidence interval
variables
coefficients
std. error
t (df=14)
p-value
95% lower
95% upper
Intercept
145.8173
64.9862
2.244
.0415
6.4358
285.1988
SF
0.0582
0.0237
2.456
.0277
0.0074
0.1090
Age
2.3687
2.0065
1.180
.2575
-1.9348
6.6722
Type
94.4710
24.8928
3.795
.0020
41.0814
147.8607
Predicted values for: Bill
95% Confidence Interval
95% Prediction Interval
SF
Age
Type
Predicted
lower
upper
lower
upper
2,700
10
1
421.053
379.816
462.289
316.377
525.729
ANOVA table
Source
SS
df
MS
F
p-value
Regression
83,886.2096
3
27,962.0699
13.90
.0002
Residual
28,171.5681
14
2,012.2549
Total
112,057.7778
17
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