Review Solutions — Bivariate Data and Scatter Plots

1. Correlation basics. Fluency

   • (a) r=0.91: Strong positive linear.
   • (b) r=−0.35: Weak negative.
   • (c) Stronger: r = −0.78.
   • (d) r=0 means unrelated?: No — no linear relationship; non-linear may exist.

2. Lines of best fit. Fluency

   • (a) ŷ=15+4(6): 39.
   • (b) Through (2,20) and (8,44): ŷ = 12 + 4x.
   • (c) Slope −3 meaning: y decreases by 3 for each unit increase in x.
   • (d) Actual 52, predicted 47: +5. Above the line.

3. r² and interpretation. Fluency

   • (a) r=0.9, r²: r² = 0.81. 81% of variation in y explained by x.
   • (b) r²=0.36, find |r|: 0.6.
   • (c) r=−0.7, unexplained %: 51%.
   • (d) r²=0.02, good predictor?: No — only 2% explained.

4. Interpolation, extrapolation, causation. Fluency

   • (a) x=18, range 5–25: Interpolation. Reliable.
   • (b) x=40: Extrapolation. Unreliable.
   • (c) Shoe size vs maths ability, ages 6–14: Age is the lurking variable.
   • (d) High r with plausible causation: e.g. altitude and temperature. Many valid answers.

5. Advertising vs sales scatter plot. Understanding

   • (a) Correlation: Strong positive linear.
   • (b) Equation: ŷ ≈ 60 + 14x.
   • (c) r=0.98, r² in context: r²=0.96. 96% of variation in sales explained by advertising spend.
   • (d) Causation comment: Causation is plausible but other factors (seasonality, competition) also play a role. A controlled experiment would confirm.

6. Two-way table (phone use vs sleep). Understanding

   • (a) High users, good sleep %: 20%.
   • (b) Low users, good sleep %: 65%.
   • (c) Association?: Yes — strong negative association between high phone use and good sleep.
   • (d) Causation? Better study design?: Not from observational data. A randomised controlled trial (randomly assigning phone limits) would help.

7. Pages vs reading time (ŷ=3.5+0.12x). Understanding

   • (a) 300-page book: 39.5 hours.
   • (b) Slope meaning: Each extra page adds ~0.12 hours (~7 min) reading time.
   • (c) 700-page book: 87.5 hrs. Extrapolation — not reliable.
   • (d) Residual for 250-page reader (40 hrs): ŷ=33.5. Residual = +6.5 hours. Slower reader than predicted.

8. Outlier investigation. Understanding

   • (a) Outlier weakens r from 0.94 to 0.82: Outlier weakened the correlation. Unusual financial situation.
   • (b) Small sample impact: With n=4, each point has large influence on r and the line.
   • (c) Data entry error: Correct the error and rerun the analysis.
   • (d) When to keep outlier: When it is a genuine, correctly recorded observation. Report analyses with and without it.

9. Rainfall vs wheat yield (r=0.87). Problem Solving

   • (a) r² and variation explained: r²=0.757. 75.7% of variation in yield explained by rainfall.
   • (b) 450 mm prediction: 23.7 t/ha.
   • (c) 800 mm — two reliability concerns: Extrapolation beyond data range; flooding may reverse the trend.
   • (d) Other variables affecting wheat yield: Temperature, soil quality, fertiliser, pests, sunlight.

10. Class size vs test score (r=−0.68). Problem Solving

    • (a) Variables: Explanatory: class size. Response: average test score.
    • (b) Slope meaning: Each extra student is associated with a 0.4-point decrease in average score.
    • (c) 30→20 students, predicted change: 73 → 77. Predicted +4 marks. But observational — actual change may differ.
    • (d) Two confounders: Wealthier schools have smaller classes and better resources. Student socioeconomic status correlates with both.

11. Mean point and line equation (perfect linear data). Problem Solving

    • (a) ¯x and ¯y: ¯x = 7, ¯y = 38.
    • (b) Slope estimate: 4 (8 increase per 2 units of x).
    • (c) y-intercept: a = 10.
    • (d) Verify all 5 points; value of r: All fit exactly. r = 1 (perfect positive linear).

12. Full bivariate report (exercise vs heart rate). Problem Solving

    • (a) Full description: Strong negative linear. r=−0.79, r²=0.624. 62.4% of variation in heart rate explained by exercise hours.
    • (b) Predict at 5 hrs: 73.5 bpm.
    • (c) Predict at 15 hrs: 50.5 bpm. Extrapolation — unreliable; also unrealistically low.
    • (d) Caution statement: Observational study — cannot prove causation. Diet, genetics, and other factors are confounders. Randomised trials needed.