L28 Solutions — Scatter Plots and Correlation

1. Describing correlation. Fluency

   • (a) Strong positive linear correlation.
   • (b) Negative; strong (|r| = 0.85 > 0.7).
   • (c) Weak positive linear correlation.
   • (d) −0.8 (|−0.8| = 0.8 is the largest).

2. Explanatory and response variables. Fluency

   • (a) Study hours (x-axis, explanatory).
   • (b) Rainfall goes on the x-axis (explanatory). Crop yield is the response.
   • (c) Neither is explanatory — shoe size and IQ have no plausible causal link. Any correlation would be spurious.
   • (d) Resale value ($) — it responds to age.

3. Reading scatter plot features. Fluency

   • (a) 15 observations.
   • (b) Non-linear.
   • (c) Outlier.
   • (d) No — r = 0 means no linear relationship, but a non-linear relationship may still exist (e.g. quadratic).

4. Estimating r from description. Fluency

   • (a) r = 0.5 (moderate positive).
   • (b) r ≈ −0.85 (strong negative).
   • (c) r ≈ −0.55 (moderate negative).
   • (d) r ≈ 0 (no correlation expected).

5. Analysing scatter plot (temperature vs beach visitors). Understanding

   • (a) Positive, linear, strong. As temperature increases, beach visitors increase in an approximately straight-line pattern.
   • (b) Approximately r ≈ 0.97 (points cluster tightly around an upward line).
   • (c) The point at (30°C, ≈480 visitors) is the outlier — despite the highest temperature, visitor numbers were low. Possible reason: rain, public holiday, or beach closure.
   • (d) No. The scatter plot shows association, not causation. Higher temperatures are associated with more visitors, but other factors (school holidays, day of week, weather besides temperature) also influence attendance.

6. Correlation vs causation. Understanding

   • (a) No. Larger fires require more trucks and cause more damage. The lurking variable is fire size.
   • (b) Age is the lurking variable: older children have bigger feet and can read better.
   • (c) Many examples, e.g. ice cream sales and drowning rates (both increase in summer); number of TVs per country and life expectancy (both relate to wealth).
   • (d) It would be unethical to randomly assign people to smoke for years. We rely on observational studies and try to control for confounders statistically.

7. Study hours vs test score. Understanding

   • (a) Strong positive — more study hours associated with higher scores.
   • (b) The student who studied 3 hours and scored 58 — slightly lower than expected; or judge by how far each is from the trend. No dramatic outlier here.
   • (c) Interpolating between 4 hrs (70) and 5 hrs (74): approximately 72.
   • (d) No — 15 hours is well outside the range of the data (1–7 hours). This is extrapolation and may be unreliable. The relationship may not continue linearly (scores are capped at 100%).

8. Comparing scatter plots. Understanding

   • (a) Scatter plot A (r = 0.92 > 0.45).
   • (b) Strong negative linear relationship — as x increases, y decreases closely following a line.
   • (c) r measures only linear correlation. A curved pattern with r = 0.7 can be misrepresented — r may underestimate the true (non-linear) relationship strength.
   • (d) No significant linear relationship between birthdate and income. r ≈ 0 ⇒ birthdate is not a useful predictor of income.

9. Fitness study (steps vs BMI). Problem Solving

   • (a) Strong negative — as daily steps increase, BMI decreases consistently.
   • (b) Explanatory: daily steps. Response: BMI.
   • (c) Cannot conclude causation. People who walk more may also eat healthier, have less sedentary jobs, or have different genetics. Confounders include diet, age, and overall activity level.
   • (d) The points follow a very consistent pattern. r is closest to −0.98 (very strong negative).

10. Misleading correlations (dentists and tooth decay). Problem Solving

    • (a) Negative correlation (more dentists associated with less tooth decay).
    • (b) Wealth / socioeconomic status: wealthier areas can afford more dentists and residents have better access to fluoride toothpaste, better diets, and regular check-ups.
    • (c) Randomly assign towns to receive additional dentists (treatment) vs no extra dentists (control), hold all other factors constant, and measure tooth decay rates after several years. This removes confounding.
    • (d) Examples: (1) Countries with more TVs per person have higher life expectancy (both linked to wealth). (2) Ice cream sales and shark attacks both rise in summer (both caused by warm weather).