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The following two questions are based on the following information:

We run a multiple regression on a sample with 500 data points.

When we include 4 independent variables, we get the following values: R2 = 0.81, SSE = 114,000.

When we add another independent variable, the results improve to: R2 = 0.90, SSE = 110,000.


What is the Adjusted R-square with 4 and 5 variables, respectively? Based on these values, which model do you recommend?

(Use at least 4 decimal places.)


To calculate the **Adjusted R-Squared** (\(R^2_{\text{adj}}\)) for both models with 4 and 5 independent variables, we can use the following formula:

\[
R^2_{\text{adj}} = 1 - \left( \frac{(1 - R^2)(n - 1)}{n - k - 1} \right)
\]

Where:
- \( R^2 \) is the coefficient of determination for the model.
- \( n \) is the number of observations (500 data points in this case).
- \( k \) is the number of independent variables.

---

### For the 4-variable model:
- \( R^2 = 0.81 \)
- \( n = 500 \)
- \( k = 4 \)

\[
R^2_{\text{adj}} = 1 - \left( \frac{(1 - 0.81)(500 - 1)}{500 - 4 - 1} \right)
\]

### For the 5-variable model:
- \( R^2 = 0.90 \)
- \( n = 500 \)
- \( k = 5 \)

\[
R^2_{\text{adj}} = 1 - \left( \frac{(1 - 0.90)(500 - 1)}{500 - 5 - 1} \right)
\]

I'll now calculate the exact values for both.The **Adjusted R-Squared** values are:

- For the 4-variable model: \( R^2_{\text{adj}} = 0.8085 \)
- For the 5-variable model: \( R^2_{\text{adj}} = 0.8990 \)

### Recommendation:
The 5-variable model has a higher adjusted \( R^2 \) value, indicating that it better explains the variance in the dependent variable when accounting for the number of predictors. Therefore, I would recommend using the 5-variable model.

Let me know if you want further details or explanations. Here are some follow-up questions:

1. What is the significance of using Adjusted \( R^2 \) instead of \( R^2 \)?
2. How does the Sum of Squared Errors (SSE) influence model selection?
3. Can you explain why adding variables increases \( R^2 \) but not always Adjusted \( R^2 \)?
4. What other criteria can be used to compare regression models?
5. What role does multicollinearity play in regression analysis?

**Tip**: Always prefer models with higher adjusted \( R^2 \), as it compensates for the number of predictors and avoids overfitting.

We run a multiple regression on a sample with 500 data points. When we include 4 independent variables, we get the following values: R2 = 0.81, SSE = 114,000. When we add another independent variable, the results improve to: R2 = 0.90, SSE = 110,000. What is the Adjusted R-square with 4 and 5 variables, respectively? Based on these values, which model do you recommend? (Use at least 4 decimal places.)

Learn how to calculate the Adjusted R-squared for a multiple regression model with 4 and 5 variables. This problem involves a sample size of 500, initial R-squared values of 0.81 and 0.90, and Sum of Squared Errors (SSE) of 114,000 and 110,000, respectively. Understand the difference between R^2 and Adjusted R^2 and how to choose the best model.

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