Math Problem Statement
Consider the following data regarding students' college GPAs and high school GPAs. The estimated regression equation is
Estimated College GPA=4.03+(−0.2380)(High School GPA).
GPAs College GPA High School GPA 2.54 4.98 3.88 3.35 2.79 3.40 2.50 2.36 3.78 3.54 3.99 2.21 Step 1 of 3 : Compute the sum of squared errors (SSE) for the model. Round your answer to four decimal places
Solution
To calculate the sum of squared errors (SSE) for the regression model, we need to follow these steps:
Step 1: Compute the predicted College GPAs.
The estimated regression equation is:
Where:
- is the predicted College GPA.
- is the High School GPA.
For each pair of High School GPA () and College GPA (), we can compute the predicted College GPA ().
Step 2: Calculate the residuals.
The residuals are the differences between the observed College GPA () and the predicted College GPA ():
Step 3: Square the residuals.
For each pair, square the residuals to get the squared errors:
Step 4: Compute the sum of squared errors (SSE).
Sum all the squared errors:
Now, let's calculate the predicted College GPAs, the residuals, and the SSE for the given data.
High School GPA () | College GPA () | Predicted College GPA () | Residual () | Squared Error () |
---|---|---|---|---|
4.98 | 2.54 | Residual Squared | ||
3.35 | 3.88 | Residual Squared | ||
3.40 | 2.79 | Residual Squared | ||
2.36 | 2.50 | Residual Squared | ||
3.54 | 3.78 | Residual Squared | ||
2.21 | 3.99 | Residual Squared |
Let's now compute the values for each of these steps. Given data high_school_gpa = [4.98, 3.35, 3.40, 2.36, 3.54, 2.21] college_gpa = [2.54, 3.88, 2.79, 2.50, 3.78, 3.99]
Regression equation parameters
intercept = 4.03 slope = -0.2380
Step 1: Compute predicted College GPA (Y_hat) for each data point
predicted_college_gpa = [intercept + slope * x for x in high_school_gpa]
Step 2: Calculate residuals (Y - Y_hat)
residuals = [y - y_hat for y, y_hat in zip(college_gpa, predicted_college_gpa)]
Step 3: Compute squared errors (residuals squared)
squared_errors = [residual ** 2 for residual in residuals]
Step 4: Sum of squared errors (SSE)
sse = sum(squared_errors) sseThe sum of squared errors (SSE) for the regression model is 2.2224 (rounded to four decimal places).
Let me know if you'd like a breakdown of any specific steps or further clarification! Here are some related questions for you:
- What does the sum of squared errors (SSE) represent in a regression model?
- How does the SSE help assess the quality of the regression model?
- What would happen if the SSE were very large in this scenario?
- Can the SSE be used to compare different regression models?
- How do you compute the total sum of squares (SST) and residual sum of squares (SSR)?
Tip: Always check the residual plots for patterns! If the residuals show a clear pattern (e.g., increasing or decreasing trends), the model might not be fully capturing the underlying data structure.
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Error Calculation
Sum of Squared Errors (SSE)
Formulas
Estimated College GPA = 4.03 + (-0.2380) * High School GPA
Residual = Y - Y_hat
Squared Error = (Y - Y_hat)^2
SSE = Σ (Y_i - Y_hat_i)^2
Theorems
Least Squares Method
Regression Analysis
Suitable Grade Level
Grades 10-12
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