Question

In Chapter 7 's Exercise 33 we examined the relationship between the fuel economy \((\mathrm{mpg})\) and horsepower for 15 models of cars. Further analysis produces the regression model \(\widehat{m p g}=46.87-0.084 H P\). If the car you are thinking of buying has a 200-horsepower engine, what does this model suggest your gas mileage would be?

Short answer

The model predicts a gas mileage of 30.07 mpg.

Step-by-step solution

Understand the Regression Model

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

Plug in the Horsepower Value

Substitute the given horsepower value of 200 into the regression model. This means replacing \( HP \) with 200 in the equation \( \widehat{mpg} = 46.87 - 0.084 \times 200 \).

Calculate the MPG

Perform the multiplication first: \(-0.084 \times 200 = -16.8\). Then, add this value to 46.87 to find \( \widehat{mpg} = 46.87 - 16.8 \).

Solve for Predicted MPG

Subtract 16.8 from 46.87 to get \( \widehat{mpg} = 30.07 \). This is the predicted fuel economy for a car with a 200-horsepower engine.

Key concepts

Fuel Economy

Fuel economy refers to how far a vehicle can travel on a specific amount of fuel. It is usually measured in miles per gallon (mpg) in countries like the United States. This means that if a car has a good fuel economy, it can travel more miles with less fuel, saving you money and reducing environmental impact.
Understanding your vehicle's fuel economy is crucial for budgeting both fuel expenses and maintenance costs. Many people choose vehicles with higher fuel economy to get the most out of every gallon they purchase.
Some factors affecting fuel economy include:

• Driving habits, such as speeding or frequent acceleration, which can make you use more fuel.
• Vehicle maintenance, as properly maintained vehicles run more efficiently.
• External conditions like terrain and weather.

Each of these factors can impact the efficiency of fuel usage, but keep in mind that the vehicle’s horsepower, as described in the original problem, is another key variable.

Horsepower

Horsepower is a unit of measurement that refers to the amount of power that an engine can produce. It is often used as an indicator of a car’s performance capabilities, particularly in terms of speed and acceleration.
Higher horsepower means the engine can perform more work in a shorter amount of time, typically leading to faster speeds. However, more horsepower often requires more fuel consumption, which can affect fuel economy.

In the context of predictive modeling and regression analysis, like in our exercise, horsepower is the independent variable. We use it to predict another variable, such as fuel economy.

• This relationship shows an inverse trend: as horsepower increases, fuel economy generally decreases.
• This happens because more powerful engines typically consume more fuel to deliver higher performance.

Understanding how horsepower impacts fuel economy helps in making informed decisions when purchasing a vehicle, particularly if balancing performance with fuel efficiency is important.

Predictive Modeling

Predictive modeling is a statistical technique used to forecast outcomes based on input data. In our fuel economy exercise, predictive modeling is demonstrated through the regression equation \[ \widehat{mpg} = 46.87 - 0.084 \times HP \].This equation shows us a linear relationship between horsepower and predicted fuel economy. By substituting a specific value into this model—such as 200 horsepower in this case—one can predict the mpg of a car.

• The predictive model in the example is linear, suggesting a straight line relationship between the variables.
• The number 46.87 is the y-intercept, showing the mpg prediction when horsepower is zero.
• The coefficient -0.084 is the slope of the line, indicating how much mpg decreases for every additional unit of horsepower.

Regression models like this are common in various fields such as economics, biology, and social sciences, as they allow us to make informed predictions based on data trends and relationships. It’s essential to remember that while predictive models provide estimates, they are not absolute truths but rather approximations based on available data.