Tomás performs four different data regressions, which result in models with the following r-values and R2-values. He now needs to choose the best model to fit his data. Which value indicates the model that best fits Tomás’s data?  A.R2=0.316  B.r=−0.991  C.R2=0.964  D.r=−0.250

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.

Learn how to choose the best regression model based on R-squared and correlation coefficients. Analyze the relationship between these statistical measures and apply them to real-world data analysis. Suitable for college-level students and professionals.

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