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.

Answer 1

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