Unit 2 Lesson 5 : Least Squares Regression

Least Squares Regression

Unit 2, Topic 2.8: Least Squares Regression

Overview

This lesson covers the least-squares regression line (LSRL), which finds the best straight line by minimizing squared residuals (prediction errors from Period 4). Use technology to calculate it, then interpret slope (y change per x unit) and intercept (y when x=0). r² shows what proportion of y's variation the line explains. Assess fit with residuals and r². Context, like who the data is from, matters because it makes interpretations real (e.g., slope of 5 points per hour for students assumes steady study).

LSRL predicts y from x; high r² means good explanation, but check residuals for fit.

Assignment:

Part 1: Guided Practice Activity

Work on your own. Use the data below from 10 students (study hours x vs. test scores y). Calculate LSRL and interpret.

Data: x (Study Hours): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 y (Test Scores): 60, 65, 70, 75, 80, 85, 90, 95, 100, 105

Tasks:

1. Calculating LSRL:
   • Use a calculator to find LSRL (LinReg: y = mx + b, e.g., m=5, b=55).
   • Write 1-2 sentences on how it minimizes squared residuals (e.g., "Line closest to points by squaring errors, reducing total distance.").
   • Extra Practice: For heights x vs. weights y (e.g., x: 160 y:50; x:170 y:60), calculate LSRL.
2. Interpreting Slope and Intercept:
   • Interpret slope in context (e.g., "Slope 5 means 5 more score points per study hour.").
   • Interpret intercept (e.g., "Intercept 55 means predicted score of 55 with 0 hours.").
   • Write 1-2 sentences assessing with context (e.g., "Intercept 55 is low but realistic for no prep; slope fits student effort.").
   • Extra Practice: Interpret slope/intercept for your heights-weights LSRL.
3. r² and Reflection:
   • Find r² (from calculator, e.g., 1.0 perfect fit).
   • Interpret as proportion (e.g., "r²=1.0 means 100% of score variation explained by hours.").
   • Write 2-3 sentences justifying model fit with residuals/r² (e.g., "High r² and random residuals confirm good fit for linear study effect.").

Part 2: Independent Practice

Use this data from 12 students: x (Sleep Hours): 5, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11 y (Focus Score): 60, 65, 70, 72, 75, 78, 80, 82, 85, 88, 90, 92

Tasks:

• Calculate LSRL 
• Interpret slope and intercept in context.
• Find r² and interpret proportion explained.
• Write 2-3 sentences assessing fit (
• Extra Activity: Invent data Calculate LSRL, r², interpret, and assess fit with context.

Homework Assignment

• Collect data from 5 people on two quantitative variables (e.g., x: coffee cups, y: productivity score). Calculate LSRL and r², interpret slope/intercept, assess fit with residuals, and justify with context to share next class.