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The problem involves calculating the Sum of Squared Errors (SSE) for the given data using the regression equation:

\[
\text{Estimated College GPA} = 2.56 + 0.1582 \times (\text{High School GPA})
\]

### Steps to compute SSE:

1. **Calculate the predicted College GPA** for each student using their High School GPA and the regression equation.
2. **Find the residuals** by subtracting the predicted College GPA from the actual College GPA for each student.
3. **Square each residual**.
4. **Sum all the squared residuals** to get the SSE.

Let's begin with calculating the predicted College GPA for each student:

- Predicted College GPA \( \hat{y}_i \) = \( 2.56 + 0.1582 \times (\text{High School GPA}) \)

Now let's calculate this for each data point:

| College GPA (Actual) | High School GPA | Predicted College GPA \( \hat{y}_i \) | Residual (Actual - Predicted) | Squared Residual |
|----------------------|-----------------|---------------------------------------|-------------------------------|------------------|
| 3.96                 | 4.42            | \( 2.56 + 0.1582 \times 4.42 \)       |                               |                  |
| 2.81                 | 3.91            | \( 2.56 + 0.1582 \times 3.91 \)       |                               |                  |
| 3.53                 | 4.21            | \( 2.56 + 0.1582 \times 4.21 \)       |                               |                  |
| 3.27                 | 2.76            | \( 2.56 + 0.1582 \times 2.76 \)       |                               |                  |
| 3.58                 | 4.95            | \( 2.56 + 0.1582 \times 4.95 \)       |                               |                  |
| 2.07                 | 4.24            | \( 2.56 + 0.1582 \times 4.24 \)       |                               |                  |

Let me calculate the predicted GPAs and the squared residuals.The Sum of Squared Errors (SSE) for the given data is approximately \( 2.1975 \) when rounded to four decimal places.

Would you like more details or have any questions on how this was calculated? 

Here are 5 related questions you could explore:

1. How is SSE related to the overall goodness of fit in a regression model?
2. What other metrics can be used to evaluate a regression model besides SSE?
3. How does the slope in a regression equation affect the prediction?
4. What is the difference between SSE and the total sum of squares (TSS)?
5. How can residual analysis help identify issues in a regression model?

**Tip:** SSE is a key metric to assess the performance of a regression model; lower SSE values indicate a better fit to the data.

Consider the following data regarding students' college GPAs and high school GPAs. The estimated regression equation is: Estimated College GPA = 2.56 + 0.1582(High School GPA). Compute the sum of squared errors (SSE) for the model.

Learn how to compute the Sum of Squared Errors (SSE) for a regression model estimating College GPAs from High School GPAs. This problem uses real data and covers concepts like residuals, predicted values, and the SSE formula.

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