Questions: Blake enters data for weight (in hundreds of pounds) and miles per gallon of cars into a statistics software package and finds a regression equation of y-hat = 38.5 - 1.4x, where weight is the explanatory variable. Based on this information, select Blake's conclusion about weight and miles per gallon that is TRUE. a.) For each additional 100 pounds of weight, miles per gallon decreases by 38.5 miles. b.) For each additional 100 pounds of weight, miles per gallon increases by 1.4 miles. c.) For each additional 100 pounds of weight, miles per gallon decreases by 1.4 miles. d.) For each additional 100 pounds of weight, miles per gallon stays relatively the same.

Blake enters data for weight (in hundreds of pounds) and miles per gallon of cars into a statistics software package and finds a regression equation of y-hat = 38.5 - 1.4x, where weight is the explanatory variable.

Based on this information, select Blake's conclusion about weight and miles per gallon that is TRUE.
a.) For each additional 100 pounds of weight, miles per gallon decreases by 38.5 miles.
b.) For each additional 100 pounds of weight, miles per gallon increases by 1.4 miles.
c.) For each additional 100 pounds of weight, miles per gallon decreases by 1.4 miles.
d.) For each additional 100 pounds of weight, miles per gallon stays relatively the same.

Transcript text: Blake enters data for weight (in hundreds of pounds) and miles per gallon of cars into a statistics software package and finds a regression equation of $\hat{y}=38.5-1.4 x$, where weight is the explanatory variable. Based on this information, select Blake's conclusion about weight and miles per gallon that is TRUE. a.) For each additional 100 pounds of weight, miles per gallon decreases by 38.5 miles. b.) For each additional 100 pounds of weight, miles per gallon increases by 1.4 miles. c.) For each additional 100 pounds of weight, miles per gallon decreases by 1.4 miles. d.) For each additional 100 pounds of weight, miles per gallon stays relatively the same.

Solution

Solution Steps

To determine the correct conclusion about the relationship between weight and miles per gallon, we need to interpret the regression equation $\hat{y}=38.5-1.4x$. The coefficient of the explanatory variable (weight) indicates the change in the dependent variable (miles per gallon) for each unit increase in the explanatory variable.

Solution Approach

Identify the coefficient of the explanatory variable (weight) in the regression equation.
Determine the effect of an additional 100 pounds of weight on miles per gallon based on the coefficient.

Step 1: Identify the Regression Equation

The given regression equation is: \[ \hat{y} = 38.5 - 1.4x \] where $ \hat{y} $ represents the predicted miles per gallon (mpg) and $ x $ represents the weight of the car in hundreds of pounds.

Step 2: Interpret the Coefficient

The coefficient of the explanatory variable (weight) is $-1.4$. This coefficient indicates the change in the dependent variable (mpg) for each unit increase in the explanatory variable (weight).

Step 3: Determine the Effect of Weight on Miles per Gallon

For each additional 100 pounds of weight ($ x $), the miles per gallon ($ \hat{y} $) decreases by 1.4 miles. This is derived directly from the coefficient $-1.4$.