The given problem provides a least-squares regression equation to model the relationship between car weights and their miles per gallon (MPG) in the city for the 2008 model year, and asks to compute the coefficient of determination $R^2$ after adding the data of a twelfth car.

The regression equation provided is: $\hat{y} = -0.005x + 38.167$ where:

• $\hat{y}$ is the predicted miles per gallon (MPG),
• $x$ is the weight of the car.

A new car is introduced with:

• Weight: 3,425 pounds,
• MPG: 13 miles per gallon.

(a) Compute the coefficient of determination $R^2$ for the expanded dataset:

The coefficient of determination $R^2$ tells us how well the regression model explains the variability in the response variable (MPG). To compute the $R^2$ for the new expanded dataset, we would need:

1. The sum of squares of residuals (SSR) from the previous 11-car dataset.
2. The total sum of squares (SST).
3. The updated SSR and SST after including the new data point for the twelfth car.

Unfortunately, without access to the original data table or the residuals from the previous dataset, we can't directly calculate the new $R^2$ in this step.

(b) Influence and outlier analysis:

To determine whether the twelfth car's data point (weight = 3,425 pounds and MPG = 13) is influential or an outlier, we would:

1. Compute the predicted MPG using the regression equation: $\hat{y} = -0.005(3425) + 38.167 = 21.0425$ The predicted MPG is approximately 21.04, but the observed MPG is 13.
2. The residual (error) for this point is: $\text{Residual} = 13 - 21.0425 = -8.0425$ Since this residual is large, it indicates that the actual MPG of the twelfth car deviates significantly from the predicted value, suggesting it might be an outlier.

To summarize:

• (a) The $R^2$ value can't be calculated without the full dataset, but adding the twelfth car may decrease $R^2$ since the new data point deviates from the model significantly.
• (b) The twelfth car appears to be an outlier due to its large residual. It could also be influential, especially if it significantly alters the regression line or the value of $R^2$.

Would you like help with further details or calculations?

Here are 5 related questions to expand on this:

1. What steps are needed to compute the sum of squares of residuals (SSR)?
2. How does adding an outlier typically affect the regression model and $R^2$?
3. What are some methods to identify influential points in a dataset?
4. How is the total sum of squares (SST) related to the coefficient of determination $R^2$?
5. Can you explain the difference between an outlier and an influential point in regression analysis?

Tip: Large residuals often indicate that a data point may be an outlier, while its influence depends on how much it affects the regression line and overall model.