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To solve the problem, we need to determine how the addition of the 12th car (with a weight of 3,425 pounds and 13 miles per gallon) affects the coefficient of determination (\(R^2\)).

### Current Regression Equation and \(R^2\)
We are given the least-squares regression line equation:
\[
\hat{y} = -0.005x + 39.324
\]
This equation suggests that as the weight of the car increases, the miles per gallon decreases.

### Step 1: Use the regression equation to estimate \( \hat{y} \) for the 12th car
For the 12th car, the weight \(x\) is 3,425 pounds. Substituting this value into the regression equation:
\[
\hat{y}_{12} = -0.005(3425) + 39.324
\]
Let's calculate this.

The predicted miles per gallon (\(\hat{y}_{12}\)) for the 12th car, based on the regression equation, is approximately 22.2.

### Step 2: Compare the predicted and actual values
- The actual miles per gallon for the 12th car is \( y_{12} = 13 \).
- The predicted miles per gallon is \( \hat{y}_{12} \approx 22.2 \).

Since there is a significant difference between the actual value (13 mpg) and the predicted value (22.2 mpg), the 12th car is an outlier. Outliers tend to decrease the coefficient of determination \( R^2 \), because they increase the overall error in the regression model.

### Conclusion:
The addition of the 12th car would most likely decrease the \( R^2 \) value. Therefore, the correct answer is:

**A. It decreases \(R^2\).**

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Would you like more details or further explanations on this topic?

Here are five related questions to expand your understanding:
1. What exactly is the coefficient of determination \( R^2 \)?
2. How do outliers affect the regression line and \( R^2 \)?
3. Can you explain how the residual is calculated for each data point?
4. How does \( R^2 \) change when the fit of the model improves?
5. What other statistical measures can assess model accuracy besides \( R^2 \)?

**Tip**: A lower \( R^2 \) means the model explains less variability in the data, indicating that additional factors might influence the relationship between the variables.

How does the addition of the twelfth car to the data set affect R^2?

Learn how adding a twelfth car with a weight of 3,425 pounds and a fuel efficiency of 13 miles per gallon affects the R-squared value in a regression model. This example covers the regression equation ŷ = -0.005x + 39.324 and explores how outliers can decrease R^2. Suitable for Grades 10-12.

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