Consider the following models A, B, and C:

Model A: Balance = β0 + β1Prime + ε

Model B: Balance = β0 + β1Time + β2Prime + β3Time × Prime + ε

Model C: Balance = β0 + β1Prime + β2Time × Prime + ε,

The regression results obtained for the models are summarized in the following table.

Variable	Model A	Model B	Model C
Intercept	88,020 (t = 77.89)	90,269 (t = 24.35)	88,020 (t = 81.19)
Time	N/A	−148 (t = −0.64)	N/A
Prime	−18,000 (t = −11.26)	−28,493 (t = −5.36)	−26,244 (t = −6.66)
Time × Prime	N/A	662 (t = 2.03)	514 (t = 2.27)
SSE	1,532,480,000	1,369,126,091	1,381,128,299
R2R-squared	0.7254	0.7547	0.7526
Adjusted R2R-squared	0.7198	0.7388	0.7421
Note: The values of relevant test statistics are shown in parentheses below the estimated coefficients.

Suppose that at a 10% significance level, you do not reject the null hypothesis, H0: β1 = 0, when testing the individual significance of Time in Model B. Would you delete Time from Model B?

Multiple Choice

Yes, removing Time from Model B results in Model C which has a higher adjusted R2R-squared.

No, Model B has the highest R2R-squared, so it should be used for making predictions.

Yes, Time should be deleted because we could not prove its significance even for α = 0.10.

No, removing Time from Model B results in Model C which has a with lower R2R-squared.

The question asks whether Time should be deleted from Model B based on the given significance level and the regression results.

Here’s a breakdown of the relevant information:

1. **Significance of Time in Model B**:
   - The p-value associated with the Time coefficient in Model B is not significant at the 10% level (as indicated by \( t = -0.64 \)). This suggests that Time is not statistically significant at this level.

2. **Comparing Adjusted \( R^2 \) Values**:
   - **Model B** has an Adjusted \( R^2 \) of 0.7388.
   - **Model C** (which excludes Time) has an Adjusted \( R^2 \) of 0.7421.

   The adjusted \( R^2 \) for Model C is slightly higher than that for Model B. Since adjusted \( R^2 \) accounts for the number of predictors in the model and Model C has a higher adjusted \( R^2 \), this suggests that Model C may provide a better fit than Model B.

Given this information, let’s analyze the options:

1. **Yes, removing Time from Model B results in Model C, which has a higher adjusted \( R^2 \).**
   - This is correct because Model C has a higher adjusted \( R^2 \) than Model B, indicating it might be a better model after removing the insignificant variable.

2. **No, Model B has the highest \( R^2 \), so it should be used for making predictions.**
   - This is incorrect because although Model B has the highest \( R^2 \), the adjusted \( R^2 \) is lower than Model C. The adjusted \( R^2 \) is a more appropriate measure when comparing models with different numbers of predictors.

3. **Yes, Time should be deleted because we could not prove its significance even for \( \alpha = 0.10 \).**
   - This is correct because Time is not significant at the 10% level, suggesting that it doesn’t contribute significantly to the model.

4. **No, removing Time from Model B results in Model C, which has a lower \( R^2 \).**
   - This is incorrect because the question should consider adjusted \( R^2 \), not just \( R^2 \). Model C actually has a higher adjusted \( R^2 \).

**Conclusion**:
The most appropriate answers are:

1. **Yes, removing Time from Model B results in Model C which has a higher adjusted \( R^2 \).**
2. **Yes, Time should be deleted because we could not prove its significance even for \( \alpha = 0.10 \).**

Either of these options would be correct.

### Would you like more details or have any questions?

### Related Questions:

1. What is the interpretation of the coefficient for Prime in Model C?
2. How does the inclusion of interaction terms affect the interpretation of regression models?
3. What is the significance of using adjusted \( R^2 \) over regular \( R^2 \) when comparing models?
4. Why is it important to consider the significance level when testing individual predictors?
5. How does the t-statistic relate to the p-value in hypothesis testing?
6. What are the implications of multicollinearity in regression models?
7. How would you interpret the interaction term \( \text{Time} \times \text{Prime} \) in Model B?
8. What are the consequences of overfitting in regression models?

**Tip**: Always check both the \( R^2 \) and adjusted \( R^2 \) values, especially when comparing models with different numbers of predictors.

Explore whether Time should be deleted from Model B based on regression results and statistical significance. Compare adjusted R-squared values between Model B and Model C to determine the most appropriate model. Understand the implications of statistical significance and adjusted R-squared in regression analysis.

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