L30 Solutions — Interpreting Bivariate Data

1. Interpreting r². Fluency

   • (a) r² = 0.64. 64% of variation in y is explained by x through the linear model.
   • (b) r² = (−0.6)² = 0.36. No — the negative sign disappears when squared. r² = 36%.
   • (c) r = ±0.7. 49% of variation in y is explained by x (sign depends on direction).
   • (d) 75% of variation in y is not explained by x (1 − 0.25 = 0.75).

2. Two-way table (pet ownership vs exercise). Fluency

   • (a) 48/80 = 60% of pet owners exercise daily.
   • (b) 22/70 ≈ 31.4% of non-pet-owners exercise daily.
   • (c) Yes — 60% vs 31.4% is a notable difference, suggesting pet owners are more likely to exercise daily.
   • (d) No — this is an observational study. Pet owners may also be more active people generally (self-selection). A controlled experiment would be needed.

3. Full interpretation (age vs blood pressure). Fluency

   • (a) Positive, moderate-to-strong (r = 0.74 > 0.7). As age increases, blood pressure tends to increase.
   • (b) r² = 0.55. 55% of variation in blood pressure is explained by age. The remaining 45% is due to other factors (diet, weight, genetics, etc.).
   • (c) ŷ = 95 + 0.5(50) = 120 mmHg. Interpolation (50 is within 30–70). Reliable.
   • (d) No — this is an observational study. Ageing is associated with rising blood pressure, but lifestyle and genetic factors also contribute. We can suggest ageing as a contributing cause but cannot prove sole causation from correlation alone.

4. Effect of removing an outlier. Fluency

   • (a) The outlier was weakening the correlation (pulling r from 0.91 down to 0.55). The true linear relationship in the remaining data is much stronger.
   • (b) Yes — 190 cm at 50 kg is exceptionally low BMI and would appear far from the main cluster in a height-weight scatter plot.
   • (c) No — outliers should be investigated first. Remove only if due to data error or if they genuinely don't belong to the same population. Legitimate outliers reveal important information.
   • (d) The outlier at (100, 5) is far to the right; it would pull the regression line toward it, likely flattening the slope and distorting predictions for smaller x values.

5. Scatter plot with outlier (sleep vs reaction time). Understanding

   • (a) Strong negative linear — as sleep hours increase, reaction time decreases in a close-to-straight-line pattern.
   • (b) r = −0.97 ⇒ strong negative. r² = 0.94: 94% of variation in reaction time is explained by sleep hours. Only 6% is due to other factors.
   • (c) Possible reasons: the person was unwell, had a stressful night despite 6 hours sleep, consumed alcohol, or there was a data recording error.
   • (d) The outlier lies far from the linear pattern. Including it would decrease |r| (weaken the apparent linear relationship), since the outlier doesn't fit the trend.

6. Comparing two classes (study vs marks). Understanding

   • (a) Class A (r = 0.92 > 0.55).
   • (b) 84.6% of variation in marks is explained by study time in Class A. The linear model fits very well.
   • (c) ŷ = 40 + 6(5) = 70. Moderately reliable (r = 0.55, r² = 0.30 — only 30% of variation explained, so predictions carry significant uncertainty).
   • (d) Possible reasons: Class B students have greater variation in background ability, different study habits, or other factors (tutoring, prior knowledge) that affect marks independently of study time.

7. Categorical bivariate data (smoking vs lung disease). Understanding

   • (a) 90/150 = 60% of smokers developed lung disease.
   • (b) 30/150 = 20% of non-smokers developed lung disease.
   • (c) Yes — 60% vs 20% is a large difference, strongly suggesting an association between smoking and lung disease.
   • (d) Causation cannot be directly concluded from an observational study alone (confounders like other health behaviours). A longitudinal cohort study with strong controls, or meta-analysis of many studies, provides stronger evidence (as exists for smoking).

8. Critique a statistical claim. Understanding

   • (a) The claim confuses correlation with causation. r = 0.78 shows a moderate-strong association, but productive workers may simply drink more coffee — or both may be linked to demanding jobs. Drinking more coffee may not increase productivity.
   • (b) Inaccurate — r = −0.4 shows a moderate negative relationship. It is not nothing; it means as x increases, y tends to decrease. The journalist is misleading readers.
   • (c) r² = 0.95 means 95% of variation is explained, but 5% is still unexplained. Additionally, the model uses sample data — predictions for new individuals will still have uncertainty (prediction intervals are wider than r² suggests).
   • (d) Books in the home are a proxy for wealth and education level — families that already value education tend to have more books and support higher IQ development. Buying books won't replicate that environment. This is a lurking variable / causation error.

9. Real estate (ŷ = 850 − 18x, r = −0.81). Problem Solving

   • (a) r = −0.81: strong negative — houses further from the city tend to be cheaper. r² = 0.656: 65.6% of variation in price is explained by distance. The remaining 34.4% is due to other factors (size, suburb, age of house).
   • (b) ŷ = 850−18(15) = 850−270 = $580 000. Interpolation (15 km is within 2–30 km range).
   • (c) ŷ = 850−18(50) = 850−900 = −$50 000. Unreliable extrapolation — negative price is meaningless. The linear model cannot be trusted beyond the data range of 2–30 km.
   • (d) Predicted at x=10: ŷ = 850−180 = $670 000. Residual = 790−670 = +$120 000. This house sold for $120 000 above what the distance-based model predicted — other factors (size, quality, suburb) likely added value.

10. Designing a bivariate study (sugar vs diabetes). Problem Solving

    • (a) Explanatory: sugar intake (g/day). Response: risk / incidence of type 2 diabetes.
    • (b) Measure daily sugar intake via food diaries or 24-hour recall surveys, and track diabetes diagnosis over several years. Ideal sample: several hundred participants (e.g. n = 500+) to detect meaningful effects and control for variation.
    • (c) r = 0.65: moderate positive linear association between sugar intake and diabetes risk. r² = 0.42: 42% of variation in diabetes risk is explained by sugar intake. We cannot conclude causation from this correlation alone — a controlled intervention study would be needed.
    • (d) Confounders include: (1) physical activity level, (2) total caloric intake / obesity, (3) family history / genetics, (4) age, (5) other dietary factors (fibre, fat). These must be measured and controlled for.