To compute the sum of squared errors (SSE), we need to follow these steps:

Formula for SSE:

$SSE = \sum (y_i - \hat{y}_i)^2$ Where:

• $y_i$ = Actual college GPA for student $i$
• $\hat{y}_i$ = Predicted college GPA for student $i$ (from the regression equation)

Regression equation:

$\hat{y} = 2.96 + 0.093 \cdot \text{(High School GPA)}$

Data:

We have the following data points for the College GPA and High School GPA:

College GPA ($y_i$)	High School GPA
3.79	2.56
3.15	4.00
3.10	4.39
3.68	4.93
3.65	4.84
2.61	3.21

Now, let's calculate the predicted college GPA ($\hat{y}_i$) for each student using the regression equation.

Step-by-Step Calculation:

1. For the first student (High School GPA = 2.56): $\hat{y}_1 = 2.96 + 0.093 \times 2.56 = 2.96 + 0.23808 = 3.19808$ $(y_1 - \hat{y}_1)^2 = (3.79 - 3.19808)^2 = 0.59192^2 = 0.3508$

2. For the second student (High School GPA = 4.00): $\hat{y}_2 = 2.96 + 0.093 \times 4.00 = 2.96 + 0.372 = 3.332$ $(y_2 - \hat{y}_2)^2 = (3.15 - 3.332)^2 = (-0.182)^2 = 0.0331$

3. For the third student (High School GPA = 4.39): $\hat{y}_3 = 2.96 + 0.093 \times 4.39 = 2.96 + 0.40737 = 3.36737$ $(y_3 - \hat{y}_3)^2 = (3.10 - 3.36737)^2 = (-0.26737)^2 = 0.0716$

4. For the fourth student (High School GPA = 4.93): $\hat{y}_4 = 2.96 + 0.093 \times 4.93 = 2.96 + 0.45849 = 3.41849$ $(y_4 - \hat{y}_4)^2 = (3.68 - 3.41849)^2 = 0.26151^2 = 0.0683$

5. For the fifth student (High School GPA = 4.84): $\hat{y}_5 = 2.96 + 0.093 \times 4.84 = 2.96 + 0.44952 = 3.40952$ $(y_5 - \hat{y}_5)^2 = (3.65 - 3.40952)^2 = 0.24048^2 = 0.0578$

6. For the sixth student (High School GPA = 3.21): $\hat{y}_6 = 2.96 + 0.093 \times 3.21 = 2.96 + 0.29853 = 3.25853$ $(y_6 - \hat{y}_6)^2 = (2.61 - 3.25853)^2 = (-0.64853)^2 = 0.4205$

Summing the squared errors:

$SSE = 0.3508 + 0.0331 + 0.0716 + 0.0683 + 0.0578 + 0.4205 = 1.0011$

Final Answer:

The sum of squared errors (SSE) is 1.0011.

Would you like me to explain any of the steps in more detail or assist with further analysis? Here are some related questions:

1. How would the SSE change if a different regression model was used?
2. How do we interpret the meaning of SSE in terms of model accuracy?
3. What would happen to the model's predictions if the data points were more spread out?
4. How can we calculate the R-squared value from SSE?
5. What is the role of residuals in the calculation of SSE?

Tip: When interpreting SSE, a smaller value indicates a better fit of the model to the data, as it suggests less error between predicted and actual values.