Question

Cellulon, a manufacturer of home insulation, wants to develop guidelines for builders and consumers on how the thickness of the insulation in the attic of a home and the outdoor temperature affect natural gas consumption. In the laboratory, it varied the insulation thickness and temperature. A few of the findings are:

Monthly Natural Gas Consumption (cubic feet),		Thickness of Insulation (inches),		Outdoor Temperature (ºF),
y		x1		x2
30.3		6.0		47.0
23.0		12		40
28.7		8		49

On the basis of the sample results, the regression equation is:

yˆ y⁢^⁢  = 116.22 − 2.09x₁ − 1.56x₂

(Note that there is no need to run a regression. Use the given equation to answer questions).

1. How much natural gas can homeowners expect to use per month if they install 6 inches of insulation and the outdoor temperature is 47 degrees F? (Round your answer to 2 decimal places.)

2. What effect would installing 7 inches of insulation instead of 6 have on the monthly natural gas consumption (assuming the outdoor temperature remains at 47 degrees F)? (Round your answers to 2 decimal places.)

3. The regression coefficients b₁ and b₂ are negative? Is this logical?

Answer 1

Here the given model is 

Where y is  Monthly Natural Gas Consumption (cubic feet),

x1 is  Thickness of Insulation (inches), and

x2 is  Outdoor Temperature (ºF),

Q a)

The amount of   natural gas can homeowners expect to use per month if they install 6 inches of insulation and the outdoor temperature is 47 degrees F will be obtained from the model.

installing 6 inches of insulation

x1 = 6

the outdoor temperature is 47 degrees F

x2 = 47

Hence the amount of   natural gas will be

=  30.36

Hence 30.36 (cubic feet)  natural gas can homeowners expect to use per month if they install 6 inches of insulation and the outdoor temperature is 47 degrees F.

Q b)

The effect would install 7 inches of insulation instead of 6 will be

x1 = 7

=  28.27

The natural gas consumption per month will decrease after installing 7 inches of insulation instead of 6

Q c)

From the equation  b1 and b2 coefficient are negative and it make sense.

Since b1 is negative and its value is -2.09 , the coefficient indicates that for every additional inch of insulation the monthly natural gas consumption will decrease by 2.09 cubic feet. Similarly  for every substracted inch of insulation, the monthly natural gas consumption will be increased by 2.09 cubic feet.

Since b2 is negative and its value is -1.56, it indicates that for every degree F increase in outdoor temperature will decrease the monthly natural gas consumption 1.56 cubic feet. Similarly   for every degrees F decrease in outdoor temperature will increase the monthly natural gas consumption 1.56 cubic feet.

( comment if needed more explanation )