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Chapter 3: Problem 37

Gas mileage We expect a car's highway gas mileage to be related to its city gas mileage. Data for all 1198 vehicles in the government's 2008 Fuel Economy Guide give the regression line predicted highway mpg \(=4.62+1.109\) (city mpg). (a) What's the slope of this line? Interpret this value in context. (b) What's the intercept? Explain why the value of the intercept is not statistically meaningful. (c) Find the predicted highway mileage for a car that gets 16 miles per gallon in the city. Do the same for a car with city mileage 28 \(\mathrm{mpg.}\)

Short Answer

Expert verified
(a) Slope is 1.109, meaning highway mileage increases by 1.109 mpg for every 1 mpg city increase. (b) Intercept is 4.62, not meaningful because 0 mpg city is unrealistic. (c) Highway mileage is approximately 22.364 mpg for 16 mpg city and 35.672 mpg for 28 mpg city.

Step by step solution

01

Identify the Slope

The slope of the regression line is the coefficient of 'city mpg' in the equation: \(1.109\). This means that for every 1 mpg increase in city mileage, the highway mileage is expected to increase by 1.109 mpg.
02

Examine the Intercept

The intercept of the regression line is \(4.62\). In context, the intercept would represent the predicted highway mpg when city mpg is 0. However, since a car cannot realistically have 0 mpg for city driving, this value is not statistically meaningful.
03

Calculate Predicted Highway Mileage for 16 mpg City

Use the regression equation to find the highway mileage when city mpg = 16: \[ \text{predicted highway mpg} = 4.62 + 1.109 \times 16 \] Calculating gives you: \[ 4.62 + 17.744 = 22.364 \] So, the predicted highway mileage is approximately 22.364 mpg.
04

Calculate Predicted Highway Mileage for 28 mpg City

Use the regression equation to find the highway mileage when city mpg = 28: \[ \text{predicted highway mpg} = 4.62 + 1.109 \times 28 \] Calculating gives you: \[ 4.62 + 31.052 = 35.672 \] Therefore, the predicted highway mileage is approximately 35.672 mpg.

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Key Concepts

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

Slope Interpretation
In the context of regression analysis, the **slope** is a crucial part of understanding how two variables are related. In our exercise, the slope of the regression line, represented by 1.109, indicates the expected change in highway mileage for each additional mile per gallon (mpg) that a car achieves in city driving.
The slope value of 1.109 means that for every 1 mpg increase in city mileage, the highway mileage is expected to increase by approximately 1.109 mpg. This reflects a positive relationship between city and highway mileage.
When interpreting the slope, it's essential to remember:
  • A positive slope, like in this case, implies that as city mileage increases, so does highway mileage.
  • If the slope were negative, it would suggest that higher city mileage correlates with lower highway mileage, which is not the case here.
  • The slope offers a numerical measure of the strength of this relationship per unit increase in city mpg.
Intercept in Context
The **intercept** in a regression line is the starting point of our predicted values when the independent variable is zero. In this exercise, the intercept is given as 4.62. It symbolizes the predicted highway mpg when city mpg is zero.
However, interpreting the intercept must be done carefully. Here, it might seem like the expected highway mileage at zero city mpg is 4.62, but this isn't practically meaningful.
Understanding the role of intercepts involves:
  • The intercept can be seen as a baseline value, but it must be scrutinized in context.
  • A zero value for city mpg isn't theoretically feasible for operating cars, making this intercept meaningless in real-world scenarios.
  • This serves as a reminder that not all calculated statistical outputs have practical significance, especially when based on values that don't occur in practical application.
Predicted Values Using Regression
Prediction is one of the primary utilities of regression analysis. With the provided equation, we can forecast highway mileage based on any given city mileage. Let's see how this works.
Suppose a vehicle's city mileage is 16 mpg. Using our regression equation:\[ \text{predicted highway mpg} = 4.62 + 1.109 \times 16 \]Calculating gives us:\[ 4.62 + 17.744 = 22.364 \]So, the predicted highway mileage is approximately 22.364 mpg. This shows how we can utilize information about one variable to predict another.
  • This process can be repeated for different city mileage values to see how the predicted highway mileage changes.
  • For example, for a city mileage of 28 mpg, calculation would yield a highway mileage of approximately 35.672 mpg.
  • These calculations demonstrate the practical application of a regression equation in making informed predictions.

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