Math Problem Statement
Consider the following data regarding students' college GPAs and high school GPAs. The estimated regression equation is
Estimated College GPA=2.91+0.1998(High School GPA).
GPAs College GPA High School GPA 3.13 2.65 3.99 2.37 3.64 2.89 3.98 4.51 3.34 2.88 3.04 2.96 Step 2 of 3 : Compute the mean square error (s2e ) for the model. Round your answer to four decimal places.
Solution
To calculate the mean square error () for the regression model, we need to follow these steps:
Step 1: Calculate the residuals
The residuals () are the differences between the observed values of College GPA and the predicted values from the regression equation. The formula for residuals is:
Where:
- is the observed value of College GPA.
- is the predicted value of College GPA, which we can calculate using the regression equation:
Step 2: Compute the residual sum of squares (RSS)
The residual sum of squares is calculated as the sum of squared residuals:
Step 3: Calculate the mean square error (MSE)
The mean square error is given by:
Where:
- is the number of data points (here, ).
- We subtract 2 from because we lose two degrees of freedom due to estimating two parameters (slope and intercept) in the regression model.
Given data:
- College GPA: [3.13, 2.65, 3.99, 2.37, 3.64, 2.89]
- High School GPA: [2.65, 2.37, 2.89, 4.51, 2.88, 2.96]
Now, let's calculate the predicted College GPA for each data point, compute the residuals, and finally the mean square error.
I'll proceed with the calculations. numpy as np
Given data
high_school_gpa = np.array([2.65, 2.37, 2.89, 4.51, 2.88, 2.96]) college_gpa = np.array([3.13, 2.65, 3.99, 2.37, 3.64, 2.89])
Regression equation coefficients
intercept = 2.91 slope = 0.1998
Predicted College GPA
predicted_college_gpa = intercept + slope * high_school_gpa
Residuals (difference between observed and predicted College GPA)
residuals = college_gpa - predicted_college_gpa
Residual sum of squares (RSS)
rss = np.sum(residuals**2)
Mean square error (MSE)
n = len(college_gpa) mse = rss / (n - 2)
Round the MSE to 4 decimal places
mse_rounded = round(mse, 4) mse_roundedThe mean square error (MSE) for the regression model is 0.8402 (rounded to four decimal places).
Would you like more details on how this was calculated or have any questions about the process? Here are some related questions that could help expand your understanding:
- What is the purpose of the residual sum of squares (RSS) in regression analysis?
- How does the degree of freedom (n - 2) affect the MSE calculation?
- What other diagnostic measures can be used to evaluate the fit of a regression model?
- How would the MSE change if we had more data points?
- What does the MSE value tell us about the accuracy of the regression model?
Tip: The closer the MSE is to 0, the better the model fits the data. A high MSE means the model's predictions are farther from the actual values.
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Math Problem Analysis
Mathematical Concepts
Regression Analysis
Residuals
Mean Square Error (MSE)
Formulas
Estimated College GPA = 2.91 + 0.1998 * (High School GPA)
Residuals (e_i) = Y_i - Y_hat_i
Residual Sum of Squares (RSS) = Σ e_i^2
Mean Square Error (MSE) = RSS / (n - 2)
Theorems
Least Squares Regression
Suitable Grade Level
Grades 11-12
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