The standard error of the model ($s_e$) is computed using the formula:

$s_e = \sqrt{\frac{\sum (Y_i - \hat{Y}_i)^2}{n-2}}$

where:

• $Y_i$ are the observed College GPAs,
• $\hat{Y}_i$ are the predicted College GPAs from the regression equation,
• $n$ is the number of data points.

Steps:

1. Compute predicted College GPAs using the regression equation:

   $\hat{Y} = 4.28 - 0.2134 \times (\text{High School GPA})$

2. Find residuals ($Y_i - \hat{Y}_i$) for each data point.

3. Square each residual and sum them.

4. Divide the sum by $n - 2$ (degrees of freedom).

5. Take the square root to get $s_e$.

I'll now calculate it. numpy as np

Given data

high_school_gpa = np.array([3.86, 3.15, 3.95, 2.27, 2.86, 4.02, 3.42, 3.08, 3.94, 4.71, 3.00, 4.51]) college_gpa = np.array([3.86, 3.15, 3.95, 2.27, 2.86, 4.02, 3.42, 3.08, 3.94, 4.71, 3.00, 4.51])

Regression equation parameters

b0 = 4.28 # Intercept b1 = -0.2134 # Slope

Compute predicted College GPAs

predicted_gpa = b0 + b1 * high_school_gpa

Compute residuals (observed - predicted)

residuals = college_gpa - predicted_gpa

Compute standard error

n = len(college_gpa) se = np.sqrt(np.sum(residuals**2) / (n - 2))

Round to four decimal places

se_rounded = round(se, 4) se_roundedThe standard error ($s_e$) of the model is 0.9181 (rounded to four decimal places).

Would you like a step-by-step breakdown of the calculations?

Here are five related questions to explore further:

1. How do we interpret the standard error in regression analysis?
2. What is the significance of the regression coefficients in this model?
3. How would the standard error change if we had more data points?
4. What other metrics can assess the accuracy of a regression model?
5. How does the coefficient of determination ($R^2$) relate to standard error?

Tip: The smaller the standard error, the better the model's predictions tend to be!