To calculate the Sum of Squared Errors (SSE) for the regression model, we follow these steps:

Steps:

1. Calculate the predicted college GPA using the regression equation: $\hat{Y}_i = 1.85 + 0.4743 \times X_i$ where $X_i$ is the high school GPA of the student.

2. Compute the residual (error) for each data point, which is the difference between the observed college GPA ($Y_i$) and the predicted college GPA ($\hat{Y}_i$): $e_i = Y_i - \hat{Y}_i$

3. Square each residual and then sum them to get the SSE: $SSE = \sum_{i=1}^{n} e_i^2$

Data:

College GPA ($Y_i$)	High School GPA ($X_i$)
3.84	2.56
3.57	3.90
2.07	3.14
4.00	3.22
3.87	2.88
2.21	2.08

Now, let's compute the predicted college GPAs using the regression equation.

We will substitute each high school GPA value ($X_i$) into the regression equation to get $\hat{Y}_i$.

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Let's begin calculating the predictions for each student. Given data high_school_gpas = [2.56, 3.90, 3.14, 3.22, 2.88, 2.08] college_gpas = [3.84, 3.57, 2.07, 4.00, 3.87, 2.21]

Regression equation constants

intercept = 1.85 slope = 0.4743

Calculate predicted college GPAs

predicted_college_gpas = [intercept + slope * x for x in high_school_gpas]

Calculate the residuals (errors)

errors = [y - y_hat for y, y_hat in zip(college_gpas, predicted_college_gpas)]

Calculate SSE (Sum of Squared Errors)

sse = sum(e**2 for e in errors) predicted_college_gpas, errors, sseHere are the results of the calculations:

Predicted College GPAs:

1. For High School GPA = 2.56, Predicted College GPA = 3.0642
2. For High School GPA = 3.90, Predicted College GPA = 3.6998
3. For High School GPA = 3.14, Predicted College GPA = 3.3393
4. For High School GPA = 3.22, Predicted College GPA = 3.3772
5. For High School GPA = 2.88, Predicted College GPA = 3.2160
6. For High School GPA = 2.08, Predicted College GPA = 2.8365

Residuals (Errors):

1. $e_1 = 3.84 - 3.0642 = 0.7758$
2. $e_2 = 3.57 - 3.6998 = -0.1298$
3. $e_3 = 2.07 - 3.3393 = -1.2693$
4. $e_4 = 4.00 - 3.3772 = 0.6228$
5. $e_5 = 3.87 - 3.2160 = 0.6540$
6. $e_6 = 2.21 - 2.8365 = -0.6265$

Sum of Squared Errors (SSE):

$SSE = 3.4379$

Thus, the Sum of Squared Errors (SSE) for the model is 3.4379.

Would you like to go into more detail or have any questions about the process?

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Here are some related questions you might find interesting:

1. How does the Sum of Squared Errors relate to model accuracy?
2. What is the significance of the residuals in regression analysis?
3. How can we improve the regression model if the SSE is high?
4. What are other metrics, apart from SSE, to evaluate regression models?
5. How would you interpret the slope coefficient in this regression equation?

Tip: Always check for outliers or anomalies in data, as they can have a significant impact on the Sum of Squared Errors and overall model performance.