Problem 4 - Basic Statistics for Business and Economics with Student

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: On the basis of the sample results, the regression equation is: $$ \hat{Y}=62.65-1.86 X_{1}-0.52 X_{2} $$ 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

Answer

a. 30.69 units; b. 1.86 units less; c. Negative coefficients are logical because they show reduced consumption with more insulation and higher temperatures.

Step 1: Identify Variables

In the regression equation, \( \hat{Y} = 62.65 - 1.86 X_1 - 0.52 X_2 \), \( X_1 \) represents the thickness of the insulation in inches, and \( X_2 \) represents the outdoor temperature in degrees Fahrenheit. \( \hat{Y} \) is the predicted natural gas consumption.

Step 2: Calculate Predicted Gas Consumption for 6 inches insulation

Substitute \( X_1 = 6 \) and \( X_2 = 40 \) into the regression equation to find the natural gas consumption: \[\hat{Y} = 62.65 - 1.86(6) - 0.52(40).\] Calculate the result: \[ \hat{Y} = 62.65 - 11.16 - 20.8 = 30.69. \] Homeowners can expect to use about 30.69 units of gas.

Step 3: Determine Effect of Additional Insulation

Substitute \( X_1 = 7 \) with \( X_2 = 40 \) into the regression equation: \[ \hat{Y_{new}} = 62.65 - 1.86(7) - 0.52(40). \] Calculate the predicted gas use: \[ \hat{Y_{new}} = 62.65 - 13.02 - 20.8 = 28.83. \] The difference in consumption is \( 30.69 - 28.83 = 1.86 \) units, showing a decrease of 1.86 units.

Step 4: Analyze Negative Regression Coefficients

The coefficients \( b_1 = -1.86 \) and \( b_2 = -0.52 \) are negative, indicating that increasing insulation thickness or increasing outdoor temperature causes a reduction in gas consumption. These coefficients make logical sense because better insulation and warmer temperatures both reduce the need for heating, thus lowering gas consumption.

Key Concepts

Natural Gas Consumption

Natural gas consumption in households primarily depends on factors such as insulation levels and outdoor temperature. In colder weather, more natural gas is typically required to maintain a comfortable indoor temperature, especially if the house lacks proper insulation. The focus here is on understanding how specific variables affect gas usage. We're using a regression analysis approach to predict how much gas a household might consume based on these conditions. Regression analysis allows us to identify patterns or relationships between the dependent variable (gas consumption) and the independent variables (insulation thickness and outdoor temperature). This is useful for making informed decisions about energy efficiency in homes.

Insulation Thickness

The thickness of insulation in a home plays a crucial role in determining energy consumption levels. Insulation acts as a barrier that helps to keep the indoor environment warm during cold weather by preventing heat from escaping. The thicker the insulation, the better it retains heat, which means less heating, and thus less natural gas, is needed. In the given regression equation, it is noted that increasing insulation thickness results in reduced gas consumption. The coefficient for insulation thickness is negative, which means a unit increase in insulation thickness leads to a significant decrease in natural gas use. This highlights the importance of proper insulation for homeowners seeking to lower their energy bills.

Outdoor Temperature

Outdoor temperature is another key factor influencing natural gas consumption. When the temperature outside drops, homes typically require more heating to maintain a comfortable indoor environment, which increases gas usage. Conversely, when it's warmer outside, less heating is needed, and natural gas consumption decreases. In the regression model provided, the coefficient for outdoor temperature is also negative. This suggests that as the outside temperature rises, the predicted natural gas consumption decreases. Understanding this relationship is valuable for homeowners and builders aiming to design energy-efficient homes that minimize heating costs, especially during colder months.

Regression Coefficients

Regression coefficients are essential for interpreting the regression equation. They provide insight into how the independent variables affect the dependent variable, which in this case is natural gas consumption. The negative values of the regression coefficients for both insulation thickness and outdoor temperature indicate an inverse relationship with gas consumption. A negative coefficient for insulation thickness shows that as insulation increases, gas consumption decreases. Similarly, a negative coefficient for outdoor temperature indicates that warmer outside temperatures lead to lower gas usage. These coefficients are logical and align with the expected outcome of improved insulation and warmer temperatures reducing heating needs, thus promoting energy efficiency.