To compute the mean square error (MSE or $s^2_e$) for the regression model, we need to follow these steps:

Step 1: Calculate the Predicted College GPA ($\hat{Y}$) for each student

Using the given regression equation:

$\hat{Y} = 4.03 - 0.2380 \times (\text{High School GPA})$

We'll compute the predicted College GPA for each student.

Step 2: Compute the residuals

The residual for each student is the difference between the observed College GPA ($Y$) and the predicted College GPA ($\hat{Y}$):

$\text{Residual} = Y - \hat{Y}$

Step 3: Compute the sum of squared residuals (SSR)

The sum of squared residuals (SSR) is calculated by:

$SSR = \sum (\text{Residual})^2$

Step 4: Compute the Mean Square Error (MSE)

The formula for $s^2_e$ (or MSE) is:

$s^2_e = \frac{SSR}{n - 2}$

Where:

• $SSR$ is the sum of squared residuals
• $n$ is the number of data points (which is 6 in this case)

Let me calculate these values for you. I'll start by computing the predicted College GPAs and then follow the steps to get the MSE. numpy as np

Given data

high_school_gpas = np.array([4.98, 3.35, 3.40, 2.36, 3.54, 2.21]) # High School GPAs college_gpas = np.array([2.54, 3.88, 2.79, 2.50, 3.78, 3.99]) # College GPAs

Regression equation: Estimated College GPA = 4.03 + (-0.2380)(High School GPA)

predicted_college_gpas = 4.03 - 0.2380 * high_school_gpas

Calculate residuals: Residual = Actual College GPA - Predicted College GPA

residuals = college_gpas - predicted_college_gpas

Sum of squared residuals (SSR)

SSR = np.sum(residuals**2)

Number of data points

n = len(high_school_gpas)

Mean Square Error (MSE)

MSE = SSR / (n - 2)

MSEThe Mean Square Error (MSE) for the model is approximately 0.5556 (rounded to four decimal places).

Would you like more details or have any questions about the process? Here are some related questions to explore:

1. How do you interpret the residuals in a regression model?
2. What is the significance of the number of data points in calculating the MSE?
3. How can the coefficient of determination (R²) be computed from the MSE?
4. What are potential ways to reduce the MSE in a regression model?
5. What is the difference between residuals and errors in regression analysis?

Tip: The MSE is a key measure of the accuracy of a regression model, with lower values indicating better fit to the data.