GPAs
College GPA	High School GPA
3.20	3.28
3.95	4.75
2.89	2.22
3.68	3.79
3.94	2.74
2.37	4.90Compute the sum of squared errors (SSE) for the model. Round your answer to four decimal places.

To compute the Sum of Squared Errors (SSE), we need to follow these steps:

1. **Find the regression model**: In this case, we need to assume that a linear regression model relates High School GPA (X) to College GPA (Y). The formula for a linear regression line is:
   \[
   Y = a + bX
   \]
   where \(a\) is the y-intercept and \(b\) is the slope.

2. **Compute the slope (b) and intercept (a)** using the following formulas:
   \[
   b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2}
   \]
   \[
   a = \frac{\sum Y - b(\sum X)}{n}
   \]
   where \(n\) is the number of data points.

3. **Calculate the predicted College GPA** for each High School GPA (X) using the regression equation \( Y_{\text{pred}} = a + bX \).

4. **Calculate the squared errors** for each data point, which is:
   \[
   \text{Squared Error} = (Y_{\text{actual}} - Y_{\text{pred}})^2
   \]
   
5. **Sum the squared errors** to find the SSE.

### Let's break down the data:

| High School GPA (X) | College GPA (Y) |
|---------------------|-----------------|
| 3.28                | 3.20            |
| 4.75                | 3.95            |
| 2.22                | 2.89            |
| 3.79                | 3.68            |
| 2.74                | 3.94            |
| 4.90                | 2.37            |

Now I will compute the necessary values and the SSE for you. numpy as np

# Given data
X = np.array([3.28, 4.75, 2.22, 3.79, 2.74, 4.90])  # High School GPA
Y = np.array([3.20, 3.95, 2.89, 3.68, 3.94, 2.37])  # College GPA

# Calculate necessary sums
n = len(X)
sum_X = np.sum(X)
sum_Y = np.sum(Y)
sum_XY = np.sum(X * Y)
sum_X_squared = np.sum(X ** 2)

# Calculate slope (b) and intercept (a)
b = (n * sum_XY - sum_X * sum_Y) / (n * sum_X_squared - sum_X**2)
a = (sum_Y - b * sum_X) / n

# Predicted Y values
Y_pred = a + b * X

# Calculate squared errors
squared_errors = (Y - Y_pred) ** 2

# Sum of squared errors (SSE)
SSE = np.sum(squared_errors)
round(SSE, 4)The Sum of Squared Errors (SSE) for the model is **1.9901**.

Would you like a detailed breakdown of the steps involved or any clarifications? Here are some related questions that might help:

1. How would you compute the coefficient of determination (R²) for this regression model?
2. What is the interpretation of the slope and intercept in this context?
3. How can you perform this calculation manually without using a programming tool?
4. What other types of regression models could be applied to this data?
5. How can you evaluate if this regression model is a good fit for the data?

**Tip**: Always ensure that the data is suitable for linear regression by checking for a linear relationship visually (e.g., via a scatter plot).

Compute the sum of squared errors (SSE) for the model. Round your answer to four decimal places.

Learn how to compute the Sum of Squared Errors (SSE) for a linear regression model with real data. This example involves College GPA and High School GPA, and demonstrates how to calculate the regression line, make predictions, and sum the squared errors. The SSE is calculated to be 1.9901 for this dataset.

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