Question

The accompanying data represent the weights of various domestic cars and their gas mileages in the city. The linear correlation coefficient between the weight of a car and its miles per gallon in the city is requalsnegative 0.992. The​ least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable is Modifying Above y with caret equals negative 0.0073 x plus 45.2243. Complete parts ​(a) through​ (c) below.Click the icon to view the data table.

1. What proportion of the variability in miles per gallon is explained by the relation between weight of the car and miles per gallon?
2. The proportion of the variability in miles per gallon explained by the relation between weight of the car and miles per gallon is_____ %. (Round to one decimal place as needed.)
3. Interpret the coefficient of determination. _____% of the variance in____ is____ by the linear model.

Answer 1

Explanation:

Hello!

Considering the variables:

Y: miles per gallon consumed by a car in the city.

X: the weight of a car.

A sample of n cars was taken and both variables measured.

The correlation coefficient was calculated r= -0.992

And the regression equation ^Y= -0.0073 + 45.2243Xi

The coefficient of determination R² is a sample measurement that determines the percentage or proportion of the variability of the dependent or response variable, Y, that is explained by the independent or explanatory variable, X, in context of the estimated regression model ^Y= a + bXi

Mathematically, the coefficient of determination between two variables is equal to the square of the coefficient of correlation of the same variables.

R²= (r)²

The three questions below are about the coefficient of determination of the linear regression between the miles per gallon and the weight of the car:

1. What proportion of the variability in miles per gallon is explained by the relation between the weight of the car and miles per gallon?

R²= (r)²= (-0.992)= 0.984

2. The proportion of the variability in miles per gallon explained by the relation between the weight of the car and miles per gallon is 98.4%. (Round to one decimal place as needed.)

0.984 * 100= 98.4%

3. Interpret the coefficient of determination. 98.4% of the variance in the miles per gallon consumed by cars in the city is explained by the linear model.

I hope this helps!