A researcher found a study relating the value of a car, y, to the age of the car, x. When researchers looked at the association of x and y, they found that the coefficient of determination was r squared equals 0.158.

Select a conclusion that the researcher can make from this data.


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


B.)
The correlation coefficient, r, is 0.842.

C.)
The correlation coefficient, r, is 0.025.

D.)
About 40% of the variation in the age of the car is explained by a linear relationship with the value of the car.

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.

A researcher found a study relating the value of a car, y, to the age of the car, x. When researchers looked at the association of x and y, they found that the coefficient of determination was r squared equals 0.158. Select a conclusion that the researcher can make from this data. A.) About 16% of the variation in the value of the car is explained by a linear relationship with the age of the car. B.) The correlation coefficient, r, is 0.842. C.) The correlation coefficient, r, is 0.025. D.) About 40% of the variation in the age of the car is explained by a linear relationship with the value of the car.

Learn how to interpret the coefficient of determination (r^2 = 0.158) in a linear relationship between car age and value. This problem explains how about 16% of the variation in car value is explained by its age. Suitable for high school students in Grades 10-12.

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