The coefficient of determination, $r^2$, represents the proportion of the variance in the dependent variable $y$ (the value of the car) that can be explained by the independent variable $x$ (the age of the car) through a linear relationship. In this case, $r^2 = 0.158$, which means about 15.8% (or approximately 16%) of the variation in the value of the car is explained by the car's age.

Correct answer:

A.) About 16% of the variation in the value of the car is explained by a linear relationship with the age of the car.

Explanation of why the other answers are incorrect:

• B.) The correlation coefficient $r$ is the square root of $r^2$, so $r = \sqrt{0.158} \approx 0.398$, not 0.842.
• C.) Similarly, $r$ is approximately 0.398, not 0.025.
• D.) $r^2 = 0.158$ relates to the variation in value of the car explained by its age, not the other way around.

Would you like more details on how to interpret $r$ and $r^2$?

Related Questions:

1. How do we calculate the correlation coefficient $r$ from $r^2$?
2. Can the coefficient of determination ever be negative?
3. What does a correlation coefficient $r$ close to 0 imply?
4. How does the value of $r^2$ change as the strength of a linear relationship increases?
5. Can the coefficient of determination $r^2$ be applied to non-linear relationships?

Tip: The coefficient of determination $r^2$ gives insight into how well the regression line fits the data, with higher values indicating a better fit.