Unit 2, Lesson 6 : Analyzing Departures from Linearity and Unit Review
Analyzing Departures from Linearity and Unit Review
Unit 2, Topic 2.9: Analyzing Departures from Linearity and Unit Review
Overview
This lesson looks at when data doesn't fit a straight line (departures from linearity) in scatterplots or residual plots, like curved patterns or trends in residuals. Transformations (e.g., log of x or y) can straighten non-linear relationships. Interpret the new model in original terms and check fit. This wraps Unit 2 with a review of associations, correlations, and regressions, plus misconceptions (e.g., correlation doesn't mean causation). Context, like who the data is from, matters because it explains curves (e.g., diminishing returns in study hours for students).
Non-linear data needs fixes like logs to use linear models; review ties it all together.
Assignment:
Part 1: Guided Practice Activity
Work on your own. Use the data below from 10 students (study hours x vs. test scores y, curved pattern). Analyze departures and transform.
Data: x (Study Hours): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 y (Test Scores): 55, 65, 75, 82, 87, 90, 92, 93, 94, 95
Tasks:
- Describing Departures from Linearity:
- Sketch a scatterplot and note curved patterns or unusual features (e.g., "Upward curve flattening at high x.").
- Calculate residuals from a trial line (e.g., y=5x + 50) for 3 points and describe (e.g., "Increasing residuals show non-linear.").
- Write 1-2 sentences in context (e.g., "Curve suggests gains slow after 5 hours, like student fatigue.").
Part 2: Unit Review and Practice
Review Unit 2 with this mixed task using sleep hours x vs. focus scores y (non-linear): x: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 y: 50, 60, 70, 78, 85, 88, 90, 91, 92, 93
Tasks:
- Describe associations from scatterplot (form, direction, strength, r).
- Identify departures (e.g., curved residuals)
- Interpret new model and r²
- Write 2-3 sentences addressing a misconception