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Cellulon, a manufacturer of home insulation, wants to develop guidelines for builders and consumers regarding the effects (1) of the thickness of the insulation in the attic of a home and (2) of the outdoor temperature on natural gas consumption. In the laboratory they varied the insulation thickness and temperature. A few of the findings are: $$\begin{array}{|ccc|}\hline \begin{array}{c}\text { Monthly Natural } \\ \text { Gas Consumption } \\\\\text { (cubic feet), }\end{array} & \begin{array}{c} \text { Thickness of } \\\\\text { Insulation } \\\\\text { (inches), }\end{array} & \begin{array}{c}\text { Outdoor } \\\\\text { Temperature } \\\\\text { ( }^{\circ} \text { F), }\end{array} \\\\\hline \text { Y } & X_{1} & X_{2} \\\\\hline 30.3 & 6 & 40 \\\26.9 & 12 & 40 \\\22.1 & 8 & 49 \\\\\hline\end{array}$$ On the basis of the sample results, the regression equation is: $$Y^{\prime}=62.65-1.86 X_{1}-0.52 X_{2}$$ a. How much natural gas can homeowners expect to use per month if they install 6 inches of insulation and the outdoor temperature is 40 degrees F? b. What effect would installing 7 inches of insulation instead of 6 have on the monthly natural gas consumption (assuming the outdoor temperature remains at 40 degrees \(F\) )? c. Why are the regression coefficients \(b_{1}\) and \(b_{2}\) negative? Is this logical?

Step by step solution

01

Identify Values for Part (a)

For part (a), we need to find the expected monthly natural gas consumption when the insulation thickness \(X_1\) is 6 inches and the outdoor temperature \(X_2\) is 40 degrees F. These values are:\(X_1 = 6\) inches\(X_2 = 40\) degrees F.

02

Use the Regression Equation for Part (a)

Substitute the values from Step 1 into the regression equation to find \(Y'\).\[Y' = 62.65 - 1.86(6) - 0.52(40)\]

03

Calculate Each Term for Part (a)

Calculate each term separately:\(-1.86 \times 6 = -11.16\)\(-0.52 \times 40 = -20.8\)

04

Sum Terms for Part (a)

Add the terms calculated in Step 3 to the intercept to find \(Y'\):\[Y' = 62.65 - 11.16 - 20.8 = 30.69\] cubic feet.

05

Identify Values for Part (b)

For part (b), we determine the change in natural gas consumption by changing \(X_1\) to 7 inches, keeping \(X_2\) at 40 degrees F.

06

Calculate Change in Consumption for Part (b)

Use the regression coefficient for \(X_1\) to calculate the change in \(Y'\):\(\Delta Y' = -1.86(7 - 6) = -1.86\) cubic feet. Monthly consumption decreases by 1.86 cubic feet.

07

Explain Regression Coefficients for Part (c)

Both \(b_1 = -1.86\) and \(b_2 = -0.52\) are negative, indicating that increasing insulation thickness or outdoor temperature reduces gas consumption. This is logical because thicker insulation and warmer temperatures both reduce heat loss, thus lowering gas consumption.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Gas Consumption

Natural gas is a common energy source used in homes for heating, particularly during colder periods. The amount of natural gas consumed typically depends on various factors that can affect heating efficiency. Key among these factors are the insulation thickness in a home and the outdoor temperature. Both elements play a significant role in determining how much energy is needed to maintain a comfortable indoor temperature.
The equation provided, \(Y^{\prime}=62.65-1.86 X_{1}-0.52 X_{2}\), predicts the monthly natural gas consumption, denoted as \(Y'\). Here, \(X_1\) represents the thickness of the insulation, and \(X_2\) the outdoor temperature. The negative signs of the coefficients indicate that either increasing insulation thickness or a rise in outdoor temperature contributes to reduced natural gas consumption.
For instance, with 6 inches of insulation and an outdoor temperature of 40°F, the expected consumption can be calculated using this equation. By substituting the values into the equation, we find that the monthly natural gas consumption is approximately 30.69 cubic feet.

Insulation Thickness

Insulation thickness is an important factor in determining the energy efficiency of a home. Insulation reduces the rate of heat transfer, thereby requiring less energy to maintain a comfortable internal temperature. In the regression analysis, insulation thickness is represented by \(X_1\). As \(X_1\) increases, the negative coefficient \(-1.86\) in the equation suggests that natural gas consumption decreases.
Thicker insulation means better heat retention during winter months, which results in less reliance on heating systems using natural gas. Builders and homeowners can use this information to optimize insulation levels. For example, the decision to increase insulation from 6 inches to 7 inches would decrease monthly gas consumption by about 1.86 cubic feet, assuming other conditions remain unchanged.
This decrease results from enhanced energy efficiency, as thicker insulation reduces the amount of cold air penetrating the home. Thus, improving insulation can be a smart environmental and economical choice.

Outdoor Temperature

Outdoor temperature, represented by \(X_2\) in the regression equation, is another critical factor affecting natural gas consumption. Generally, warmer outdoor temperatures reduce the need for heating, translating into lower natural gas use. The value \(-0.52\) for the \(X_2\) coefficient indicates that each degree Fahrenheit increase in temperature results in a decrease in natural gas consumption by 0.52 cubic feet, given constant insulation.
With higher outdoor temperatures, homes naturally lose less heat to the environment, allowing for more efficient heating without relying heavily on natural gas. Seasonal changes heavily influence this factor, as evident during winter months when lower temperatures necessitate increased natural gas usage to maintain comfortable indoor climates.
Understanding the impact of outdoor temperature can help homeowners plan for seasonal fluctuations in energy bills. It emphasizes the importance of combining adequate insulation with temperature monitoring to achieve optimal heating efficiency.

Regression Coefficients

Regression coefficients \((b_1\) and \(b_2)\) are crucial in understanding the relationship between independent variables (insulation thickness and outdoor temperature) and the dependent variable (natural gas consumption). In the given equation, \(b_1 = -1.86\) and \(b_2 = -0.52\). These coefficients quantify the expected change in the monthly gas consumption for one unit change in insulation thickness and temperature, respectively.
The negative nature of both coefficients aligns logically with the context of energy consumption. As insulation thickness (\(X_1\)) increases, natural gas consumption decreases because of enhanced thermal efficiency. Similarly, as the outdoor temperature (\(X_2\)) rises, less heating is needed, leading to reduced consumption.
This relationship shows a clear, predictable pattern of energy savings, helping architects and homeowners make informed decisions about building materials and thermostat settings. Accurate interpretation of these coefficients allows for effective predictions and optimizations tailored to specific climate conditions.