To determine the model that best fits Tomás’s data, you need to consider both the $r$-value (correlation coefficient) and the $R^2$-value (coefficient of determination):

• The $r$-value indicates the strength and direction of the linear relationship between the variables. The closer the absolute value of $r$ is to 1, the stronger the correlation. A positive $r$ value indicates a positive relationship, and a negative $r$ value indicates a negative relationship.

• The $R^2$-value indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s). The closer $R^2$ is to 1, the better the model fits the data.

Options given:

• A. $R^2 = 0.316$
• B. $r = -0.991$
• C. $R^2 = 0.964$
• D. $r = -0.250$

Analysis:

• $R^2 = 0.316$ suggests a weak model fit.
• $r = -0.991$ suggests a very strong negative linear correlation, close to perfect.
• $R^2 = 0.964$ suggests a very strong model fit.
• $r = -0.250$ suggests a weak negative correlation.

Conclusion: The best indicator of a well-fitting model is a high $R^2$ value because it directly shows how well the model explains the variance in the data. Therefore, the correct answer is C. $R^2 = 0.964$.

Would you like more details or have any questions?

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Here are 5 related questions you might explore:

1. How does the $r$-value relate to the $R^2$-value?
2. Can a model have a high $r$-value but a low $R^2$-value?
3. What are the implications of a negative $r$-value for the model's predictions?
4. How do outliers affect the $R^2$ and $r$-values in regression analysis?
5. What are some limitations of using $R^2$ as the only metric for model selection?

Tip: While $R^2$ is a good indicator of model fit, always consider the context of your data and whether a high $R^2$ truly reflects meaningful predictions.