https://file.aiquickdraw.com/mathbot/file/1725859311653hrzdlh70.jpg

I see the image contains statistical output and questions related to an Ordinary Least Squares (OLS) regression model.

Let me walk through the key components:

### Regression Model:
- **Dependent variable**: GPA (Grade Point Average)
- **Independent variable**: ACT (ACT score)
- **Coefficient for ACT**: 0.031612
  - This suggests that for each unit increase in ACT score, the GPA is expected to increase by approximately 0.0316 points.
  
### Key Statistics:
- **R-squared**: 0.924141
  - This means that about 92.41% of the variability in GPA is explained by the ACT scores.
- **F-statistic**: 78.68260 (significant at p < 0.01)
  - This is a strong indicator that the model has predictive power.

Let me now address each of the questions in the image:

1. **\( \hat{\beta_1} \)**:
   - \( \hat{\beta_1} \) refers to the slope coefficient for ACT, which is 0.031612.
   
2. **Total Sum of Squares (SST)**:
   - This is a measure of the total variance in GPA. The value can be found or calculated using the formula: 
   \[
   SST = SSE + SSR
   \]
   Based on the image, you'd need to extract the Sum of Squared Residuals (SSE) and Regression Sum of Squares (SSR) to calculate this, but these values are not directly shown.

3. **How much higher is the GPA predicted to be if the ACT score is increased by 5 points?**
   - To compute this:
   \[
   \Delta \text{GPA} = 0.031612 \times 5 = 0.15806
   \]
   So, the GPA would increase by approximately 0.158 points for a 5-point increase in ACT.

4. **Predicted value of GPA when ACT = 20**:
   - The equation of the regression line is:
   \[
   \text{GPA} = \beta_0 + \beta_1 \times \text{ACT}
   \]
   With \( \beta_0 = 0.921827 \) and \( \beta_1 = 0.031612 \):
   \[
   \text{Predicted GPA} = 0.921827 + 0.031612 \times 20 = 0.921827 + 0.63224 = 1.55407
   \]
   So, the predicted GPA when ACT = 20 is approximately 1.554.

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Would you like further explanation on any part of the solution?

Here are 5 related questions that expand on this:

1. How would the model change if another independent variable (like hours studied) was added?
2. What is the interpretation of the p-value associated with the ACT score?
3. What does an R-squared value close to 1 imply about the model fit?
4. How would you calculate the residual for a specific student with known GPA and ACT?
5. Can the F-statistic be used to test if the model is useful? Why or why not?

**Tip**: Always check the p-value of coefficients to ensure they are statistically significant before interpreting their impact on the dependent variable.

Your consultancy firm has been hired by the American Council of Universities to evaluate the relationship between ACT scores and GPA. How much higher is the GPA predicted to be if the ACT score is increased by 5 points?

Explore the relationship between ACT scores and GPA using an Ordinary Least Squares (OLS) regression model. The analysis focuses on predicting GPA based on ACT score increases, including detailed interpretation of coefficients and statistical significance.

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