Question

The regression model \(\widehat{m p g}=46.87-0.084 H P\) relates cars \(^{\prime}\) horsepower to their fuel economy (in mpg). Explain what the slope means.

Short answer

The slope means that for each additional unit of horsepower, fuel economy decreases by 0.084 mpg.

Step-by-step solution

Identify the Regression Model

The given regression model is \(\widehat{mpg} = 46.87 - 0.084 \times HP\), where \(\widehat{mpg}\) represents the estimated miles per gallon and \(HP\) represents the horsepower of the car.

Understand the Slope Term

The slope ( 0.084) is a coefficient in the regression model that shows the relationship between the dependent variable (mpg) and the independent variable (HP, horsepower).

Interpret the Slope Numerically

Since the slope is \(-0.084\), this indicates that for every one unit increase in horsepower, the estimated miles per gallon decreases by 0.084 mpg, holding other factors constant.

Relate the Slope Back to the Context

In the context of this model, the negative slope signifies that as a car's horsepower increases, its expected fuel economy (measured in mpg) decreases slightly.

Key concepts

Slope Interpretation

The concept of slope in a regression line is instrumental in understanding how variables relate to each other in a model. In a linear regression equation such as \(\widehat{mpg} = 46.87 - 0.084 \times HP\), the slope is the coefficient that accompanies the independent variable, which, in this case, is horsepower (HP). The slope value here is \(-0.084\). This negative sign is crucial; it indicates the direction of the relationship. With this model, for every additional unit of horsepower, the fuel economy, as measured in miles per gallon (mpg), is expected to reduce by 0.084 mpg.

• If the slope were positive, it would indicate an increase in mpg with additional horsepower.
• The actual numerical value of the slope tells you the magnitude of this change, demonstrating how strongly HP affects mpg.

The slope helps predict the dependent variable (mpg), given a particular value of the independent variable (HP), providing insights into the trade-offs between different attributes of cars.

Linear Regression

Linear regression is a statistical technique used to model the linear relationship between a dependent variable and one or more independent variables. In this context, it helps us understand how changes in a car’s horsepower (independent variable denoted as HP) affect its fuel economy (dependent variable represented by miles per gallon, or mpg).The equation for linear regression takes the form of \(y = b_0 + b_1x\), where:

• \(y\) is the predicted value of the dependent variable.
• \(b_0\) is the y-intercept, representing the estimated mpg when horsepower is zero.
• \(b_1\) is the slope, indicating the change in mpg for each one-unit increase in horsepower.

In fuel economy modeling, linear regression simplifies the relationship into a straight line, making it easier to predict changes and make informed decisions about vehicle performance characteristics. It's an essential tool for uncovering trends and relationships in data.

Fuel Economy Modeling

Fuel economy modeling is the process of using mathematical and statistical methods to estimate how various factors influence a vehicle's fuel consumption. When it comes to cars, aspects like weight, aerodynamics, and especially horsepower play significant roles in determining fuel efficiency.In this scenario, the model \(\widehat{mpg} = 46.87 - 0.084 \times HP\) focuses specifically on horsepower as a key factor impacting fuel efficiency. By using regression analysis:

• We can estimate fuel consumption based on horsepower.
• This model helps automotive engineers, designers, and consumers understand the trade-offs involved with increasing horsepower.

Fuel economy modeling provides valuable insights into how modifications and innovations in car design and technology could lead to more efficient vehicles, ultimately supporting better decision-making in automotive development and environmental policy.