Unit 2, Lesson 4 : Linear Regression Models and Residuals

Linear Regression Models and Residuals

Unit 2, Topics 2.6–2.7: Linear Regression Models and Residuals

Overview

This lesson explains linear regression models, which predict a response variable (y, like test scores) from an explanatory variable (x, like study hours). Residuals are the errors between actual and predicted y values. Residual plots show if the line fits well (random dots = good; patterns = curved data). Context, like the group studied, helps explain errors (e.g., student residuals might reflect stress).

Linear models use the equation y = mx + b (m = slope, b = starting point) for predictions. Residuals check the 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). Define a model and calculate residuals.

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. Defining Linear Regression Models:
   • Define a linear model to predict y from x (e.g., scores = 55 + 5(hours)).
   • Use a calculator to find the line (LinReg: slope m ≈ 5, intercept b ≈ 55).
   • Write 1-2 sentences interpreting in context (e.g., "Each extra hour predicts 5 more points for students.").
   • Extra Practice: For heights x vs. weights y (e.g., x: 160, y: 50; x: 170, y: 60), define a model and interpret.
2. Calculating Residuals:
   • Calculate residuals for 3 points (e.g., predicted for x=1: 60, observed 60, residual=0; x=2: 65, observed 65, residual=0).
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   • Write 1-2 sentences interpreting as errors (e.g., "Positive residual means actual score beat prediction, like extra effort.").
   • Extra Practice: Calculate residuals for 2 points in your heights-weights data.
3. Constructing and Describing Residual Plots:
   • Sketch a residual plot (x=study hours, y=residuals; plot points, look for random scatter).
     
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   • Write 2-3 sentences describing (e.g., "Random scatter suggests good linear fit, no patterns like curves.").
   • Reversing Interpretations: For a given residual plot (random dots), create 2 interpretations (e.g., "No pattern means line fits; curved residuals would mean non-linear.").

Part 2: Independent Practice

Use this data from 12 students:

x (Exercise Min): 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120
y (Energy Score): 50, 56, 60, 64, 70, 74, 80, 86, 90, 94, 100, 106

Tasks:

• Define a linear model and Calculate the Linear Regression LinReg
• Calculate residuals for 3 points and interpret errors in context.
• Sketch a residual plot and describe linearity (e.g., "Random scatter confirms fit for exercise-energy link.").
• Write 2-3 sentences justifying a claim (e.g., "Model predicts well, but positive residuals show some overperform, like motivated students.").

Homework Assignment

• Collect data from 5 people on two quantitative variables (e.g., x: reading min, y: quiz score). Define a linear model, calculate residuals for 2 points, sketch a residual plot, describe fit, and interpret with context to share next class.