Unit 2, Lesson 3 : Representing Quantitative Relationships and Correlation

Representing Quantitative Relationships and Correlation

Unit 2, Topics 2.4–2.5: Representing Quantitative Relationships and Correlation

Overview

This lesson introduces scatterplots for two quantitative variables (like study hours and test scores) to show relationships. Identify explanatory (x-axis, cause-like) and response (y-axis, effect-like) variables. Describe form (linear/non-linear), direction (positive/negative), strength (strong/moderate/weak), and unusual features (clusters/outliers). Calculate correlation r (using technology) to measure linear strength, but note r near ±1 doesn't guarantee linearity. Context, like who the data is from, matters because it explains patterns (e.g., more study linking to higher scores in students).

Scatterplots plot points to reveal associations; r quantifies linear ones.

Assignment:

Part 1: Guided Practice Activity

Work on your own. Use the data below from 10 students (study hours vs. test scores). Represent with scatterplots and calculate r.

Data: Study Hours (x): 1, 2, 2, 3, 3, 4, 5, 5, 6, 7 Test Scores ( y ): 60, 65, 70, 68, 75, 80, 78, 85, 82, 90

Tasks:

1. Representing with Scatterplots:
   • Identify explanatory (x: study hours) and response (y: test scores).
   • Sketch a scatterplot (plot points by hand, label axes).
   • Write 1-2 sentences describing form, direction, strength, and unusual features (e.g., "Linear positive form, moderate strength, no outliers.").
   • Extra Practice: For heights (x) vs. weights ( y ) from 5 people (e.g., x: 160,165; y: 50,55), sketch a scatterplot and describe.
2. Calculating and Interpreting Correlation:
   • Use a calculator to find r (correlation coefficient).
   • Write 1-2 sentences interpreting direction/strength in context (e.g., "r=0.94 shows strong positive link; more hours predict higher scores in students.").
   • Note if r near ±1 guarantees linearity (e.g., "No, curved patterns can have high r.").
   • Extra Practice: Estimate r for your heights-weights data and interpret.
3. Quickwrite Reflection:
   • Quickwrite 2-3 sentences on a scatterplot output (e.g., from image1.png: points rising but with outlier). Assess a claim (e.g., "The outlier weakens the linear claim for student performance.").

Part 2: Independent Practice

Use this data from 12 students: Hours Slept (x): 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 10 Test Scores ( y ): 65, 70, 68, 75, 78, 80, 82, 85, 78, 88, 90, 92

Tasks:

• Sketch a scatterplot, labeling x (sleep hours) and y (scores).
• Describe form, direction, strength, and unusual features.
• Use a calculator for r, interpret in context (e.g., "r=0.75 moderate positive; more sleep links to better scores, but outlier at 78 weakens it.").
• Write 2-3 sentences assessing a claim (e.g., "Sleep predicts scores linearly, but context like stress might explain the dip.").
• Extra Activity: Invent data (e.g., x: exercise min, y: energy level). Sketch scatterplot, estimate r, and claim with context.

Homework Assignment

• Collect data from 5 people on two quantitative variables (e.g., x: coffee cups, y: alertness score). Sketch a scatterplot, describe features, estimate/calculate r, and interpret a claim with context to share next class.