[DATA] Weight of a Car versus Miles per Gallon (Refer to Problem 37, Section 4.1.) An engineer wants to determine how the weight of a car, x, affects gas mileage, y. The following data represent the weights of various domestic cars and their miles per gallon in the city for the 2015 model year.
b. Explain how to interpret the slope and y-intercept, if it makes sense to do so.

Verified step by step guidance

1. Identify the regression equation that models the relationship between the weight of a car (x) and miles per gallon (y). This equation typically has the form: \(y = \beta_0 + \beta_1 x\), where \(\beta_0\) is the y-intercept and \(\beta_1\) is the slope.
2. Interpret the slope (\(\beta_1\)): The slope represents the average change in miles per gallon (y) for each one-unit increase in the weight of the car (x). In other words, it tells us how much the gas mileage is expected to increase or decrease when the car's weight increases by one unit (usually pounds or kilograms).
3. Interpret the y-intercept (\(\beta_0\)): The y-intercept is the predicted miles per gallon when the weight of the car is zero. Consider whether this value makes sense in the context of the problem. Since a car cannot have zero weight, the y-intercept may not have a meaningful real-world interpretation but is necessary for the equation.
4. Explain that the slope's sign (positive or negative) indicates the direction of the relationship: a negative slope suggests that heavier cars tend to have lower miles per gallon, while a positive slope would suggest the opposite.
5. Summarize that the slope provides practical insight into how weight affects fuel efficiency, while the y-intercept is mainly a mathematical artifact unless zero weight is a meaningful or possible value.

Key Concepts

Slope in Linear Regression

The slope represents the rate of change in the dependent variable (miles per gallon) for each one-unit increase in the independent variable (car weight). It indicates how much gas mileage is expected to decrease or increase as the car's weight changes, assuming a linear relationship.

Y-Intercept in Linear Regression

The y-intercept is the predicted value of the dependent variable when the independent variable is zero. In this context, it represents the estimated miles per gallon for a car weighing zero pounds, which may or may not be meaningful depending on whether such a value is realistic.

Contextual Interpretation of Regression Parameters

Interpreting slope and intercept requires considering the real-world context and data range. For example, a zero car weight is not practical, so the y-intercept might lack meaningful interpretation, while the slope provides insight into how weight affects fuel efficiency within the observed data.