Solutions: Scatterplots and Correlation

Description: The association is negative (as hours of exercise increase, resting heart rate decreases), linear (the points follow a roughly straight-line pattern), and strong (points are close to the line with little scatter).

Direction: negative — the points go downward from left to right, meaning more exercise is associated with lower heart rate.

Form: linear — the points follow a roughly straight-line pattern (not curved).

Strength: strong — the points cluster closely around an imagined straight line.
Matching:
- r = 0.94 → Strong positive linear
- r = −0.61 → Moderate negative linear
- r = 0.28 → Weak positive linear
- r = −0.88 → Strong negative linear
r = 0.94: positive sign, |r| = 0.94 ≥ 0.75 → strong positive linear.

r = −0.61: negative sign, 0.5 ≤ |r| < 0.75 → moderate negative linear.

r = 0.28: positive sign, |r| < 0.5 → weak positive linear.

r = −0.88: negative sign, |r| ≥ 0.75 → strong negative linear.
(a) r = 0.83 indicates a strong, positive linear association — larger households tend to have higher electricity bills.

(b) The number of people in the household is the explanatory variable (x), as household size likely influences electricity use, not the other way around.

(c) No — correlation does not imply causation. While r = 0.83 shows a strong association, there may be other explanations. Larger households may have bigger homes (more appliances, more heating/cooling). A controlled experiment would be needed to establish causation, but this is not ethically or practically possible here.
(a) Distance from CBD and house price: likely negative (properties closer to the city tend to be more expensive, though this varies by suburb).

(b) Daily rainfall and beach visitors: negative (more rain, fewer people at the beach).

(c) Shoe size and IQ: near zero (no meaningful relationship between foot size and intelligence).

(d) Age of car and resale value: negative (older cars generally have lower resale value due to depreciation).
(a) r = 0.38 indicates a weak positive linear association between sprint speed and vertical jump height. There is some tendency for faster sprinters to also jump higher, but the relationship is not strong — sprint speed is not a reliable predictor of jump height among these athletes.

(b) Removing this outlier (fast sprint, low jump — a point that goes against the general trend) would likely increase r, making the association appear stronger, because the outlier is pulling the correlation coefficient downward.
(a) There is a strong, positive, linear association between monthly rainfall and wheat yield (r = 0.77). Months with higher rainfall tend to have higher wheat yield.

(b) The statement oversteps what correlation evidence can support. While r = 0.77 indicates a strong association, correlation does not establish causation. Other factors (soil quality, temperature, fertiliser) may also explain the pattern. The recommendation to build an irrigation system is a business decision requiring more evidence than correlation alone.

(c) To establish causation, a controlled experiment would be needed — for example, randomly assigning identical plots of land to different irrigation levels while controlling for soil, temperature, and fertiliser, then comparing yields.
(a) Pearson's r is sensitive to outliers. A single point far from the main cluster can increase or decrease r substantially, because r is based on the sum of products of deviations from the mean. If the outlier reinforces the overall trend, it increases |r|. If it goes against the trend, it decreases |r|.

(b) A statistician should always examine the scatterplot before reporting r. The scatterplot reveals the form of the association (linear vs non-linear), the presence of outliers, and whether r is an appropriate summary measure.
(a) The lurking variable is age. As children get older, both their feet (shoe size) and reading ability increase — not because these two things are directly related, but because both are consequences of growing up.

(b) Larger feet do not help children read better. Age drives both variables simultaneously. The correlation is spurious — caused by a third variable (age) rather than a direct link between shoe size and reading ability.

(c) The researcher could control for age by analysing data within specific age groups (e.g., only comparing children of the same age). Within each age group, the correlation between shoe size and reading ability would disappear or be very weak.
(a) Study hours (x) is the explanatory variable (it likely influences exam score). Score (y) is the response variable. Plot x on the horizontal axis, y on the vertical axis.

(b) The association is positive (more study hours → higher score), linear (points follow a straight-line pattern), and strong (points are close to the line).

(c) Using technology: r ≈ 0.998. This indicates an extremely strong, positive, linear association between study hours and exam score. As study hours increase by 1 hour, exam score tends to increase by approximately 6–7 marks.
(a) No. r = 0.79 shows a strong positive association at the country level (ecological correlation), but this does not prove that eating chocolate boosts intelligence or causes Nobel Prize wins. Ecological correlations (comparing countries rather than individuals) are especially prone to confounding.

(b) Possible lurking variables: (i) national wealth — wealthier countries can afford both more chocolate and better universities/research funding; (ii) general standard of living — higher GDP per capita supports better education and greater chocolate consumption independently.

(c) A randomised controlled trial would be needed — for example, randomly assigning individuals to different dietary treatments (chocolate vs control) over an extended period and measuring cognitive outcomes. This is extremely difficult to conduct properly, and the Nobel Prize outcome is not measurable in an experiment.