https://file.aiquickdraw.com/mathbot/file/1728655674341c2am235e.jpg

To determine whether the regression model overestimates or underestimates the tail length based on the residual, we can follow these steps:

### Step 1: Formulate the regression equation.
The regression equation is:

\[
\text{Predicted Tail Length} = \text{Constant} + (\text{Total Length Coefficient}) \times \text{Total Length}
\]

From the table:
- Constant = -1.281
- Total Length Coefficient = 0.5264

Thus, the regression equation becomes:

\[
\text{Predicted Tail Length} = -1.281 + 0.5264 \times \text{Total Length}
\]

### Step 2: Calculate the predicted tail length.
Given that the total length of the roadrunner is 59.0 cm, we substitute this value into the equation:

\[
\text{Predicted Tail Length} = -1.281 + 0.5264 \times 59.0
\]

\[
\text{Predicted Tail Length} = -1.281 + 31.0546 = 29.7736 \, \text{cm}
\]

### Step 3: Compare the predicted tail length to the actual tail length.
The actual tail length of the roadrunner is given as 31.1 cm. The residual is the difference between the actual and the predicted values:

\[
\text{Residual} = \text{Actual Tail Length} - \text{Predicted Tail Length} = 31.1 - 29.7736 = 1.3264 \, \text{cm}
\]

### Step 4: Determine overestimate or underestimate.
Since the residual is positive, the actual tail length is greater than the predicted tail length, meaning the model **underestimated** the tail length.

### Conclusion:
The correct answer is:

**A: Underestimate, because the residual is positive.**

Would you like a more detailed explanation or have any questions about the process? Here are 5 related questions to deepen your understanding:

1. How is the residual value used to assess model accuracy in regression?
2. What does a positive or negative residual imply in a regression model?
3. How does the coefficient of the independent variable (Total Length) affect predictions?
4. What would happen to the predicted value if the constant term were higher?
5. How do standard errors (SE Coef) in regression analysis help in understanding the accuracy of the coefficients?

**Tip**: Always check the sign of the residual to determine whether a prediction overestimates or underestimates the actual value.

Suppose a roadrunner has a total length of 59.0 cm and tail length of 31.1 cm. Based on the residual, does the regression model overestimate or underestimate the tail length of the roadrunner?

This problem involves a regression analysis predicting the tail length of a roadrunner based on its total length. Using coefficients from a linear regression model, learn how to calculate the predicted tail length, compute the residual, and determine whether the model overestimates or underestimates the actual value. Suitable for Grades 10-12.

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