Lesson 6: 3.6 The Least-Squares Regression Line
Duration of Days: 3
Lesson Objective
Students will be able to calculate and interpret residuals, and understand how the "least-squares" method minimizes the sum of squared residuals to create the line of best fit.
How do we mathematically define an "error" in our prediction?
Out of all possible lines we could draw, what makes the "Least-Squares" line the winner?
Residual
Least-squares regression line
Sum of squared residuals
Centroid
HSS-ID.B.6.B: Informally assess the fit of a function by analyzing residuals.
The SAT often asks students to identify the residual of a specific point (the vertical distance from the point to the line). Knowing that Residual = Observed - Predicted is essential.
his section moves from "any line" to the best line. It introduces the residual as the fundamental measure of accuracy.
The Problem: Using a dataset of shoe sizes and heights, the LSRL is y-hat = 105 + 6.2x. One student wears a size 10 shoe and is 170 cm tall.
Task: Calculate and interpret the residual for this student. Did the model over-predict or under-predict their height?
Residual Direction: Students often do Predicted - Observed. Use the mnemonic "AP" (Actual - Predicted) or "Observed - Predicted" to keep the order straight.
Squared residuals: Students ask why we square them. Explain that if we didn't, the positive and negative errors would cancel each other out.
Support: Use a physical "string and pin" activity on a giant graph. Have students try to minimize the "total distance" with the string to see how the line shifts.
Extension: Prove that the sum of the residuals for any LSRL is always zero.
Teacher assigns examples from the textbook and other resources.
Access E-Book through Classlink