To compute the mean square error (MSE) or s²e, we first need to follow these steps:

Step 1: Calculate the predicted College GPAs using the given regression equation.

The regression equation is:

$\hat{Y} = 0.03 + 0.9077 \times X$ where:

• $\hat{Y}$ is the predicted College GPA,
• $X$ is the High School GPA.

We'll calculate the predicted College GPAs for each High School GPA provided.

Step 2: Calculate the residuals for each student.

The residual for each student is the difference between the observed College GPA and the predicted College GPA.

$\text{Residual} = \text{Observed College GPA} - \hat{Y}$

Step 3: Calculate the sum of squared residuals (SSR).

The sum of squared residuals is calculated as:

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

Step 4: Calculate the mean square error (MSE).

The formula for MSE is:

$MSE = \frac{SSR}{n - 2}$

where:

• $n$ is the number of data points (students).

Let's compute the MSE step-by-step.

Given Data:

College GPA	High School GPA
2.08	2.12
2.27	2.65
3.97	4.09
3.07	3.55
3.46	3.46
3.14	3.76

Now, let me calculate the predicted values, residuals, and the final MSE. numpy as np

Given data

high_school_gpa = np.array([2.12, 2.65, 4.09, 3.55, 3.46, 3.76]) college_gpa = np.array([2.08, 2.27, 3.97, 3.07, 3.46, 3.14])

Regression equation: Y_hat = 0.03 + 0.9077 * X

predicted_college_gpa = 0.03 + 0.9077 * high_school_gpa

Residuals: Observed College GPA - Predicted College GPA

residuals = college_gpa - predicted_college_gpa

Sum of squared residuals (SSR)

ssr = np.sum(residuals**2)

Mean square error (MSE)

n = len(college_gpa) mse = ssr / (n - 2)

mse, ssr, predicted_college_gpa, residuals### Results:

• Predicted College GPAs based on the regression equation:

  • 1.9543, 2.4354, 3.7425, 3.2523, 3.1706, 3.4430

• Residuals (Observed College GPA - Predicted College GPA):

  • 0.1257, -0.1654, 0.2275, -0.1823, 0.2894, -0.3030

• Sum of Squared Residuals (SSR): 0.3037

• Mean Square Error (MSE): 0.0759

Thus, the MSE is 0.0759 (rounded to four decimal places).

Let me know if you'd like more details or have any questions!

Related Questions:

1. How would you calculate the coefficient of determination (R²) for this model?
2. What is the standard error of the estimate?
3. How would you interpret the slope (0.9077) in the context of this data?
4. What assumptions are made in simple linear regression models like this one?
5. How can you improve the predictive accuracy of the regression model?

Tip:

The residuals give you an indication of how well the model fits the data. A small residual means the model’s prediction was close to the actual value.