Interpreting Bivariate Data — Solutions

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1. Interpret described datasets

   1. Exercise vs energy:
      1. Adults who exercised more hours per week tended to report higher energy levels. The strong positive correlation indicates this is a consistent trend across the 40 participants.
      2. No. This is an observational study — we cannot establish causation from correlation alone. It is possible that people with higher natural energy levels choose to exercise more, or that a third variable (e.g. diet, sleep quality) influences both.
   2. Sodium vs kidney function:
      1. The sample of 8 patients is very small. With such a small sample, a few unusual results could heavily influence the correlation, making it unreliable for generalisation.
      2. No. A sample of 8 is far too small to justify a nationwide policy recommendation. The study also does not establish causation, and the specific patient group (e.g. their age, existing conditions) may not represent the general population.
   3. Employee age vs sick days:
      1. Almost no correlation means that knowing an employee’s age gives very little information about how many sick days they are likely to take. The two variables are essentially unrelated in this dataset.
      2. No. With no correlation, age is not a useful predictor of sick days. Using age to make decisions about sick leave could also be discriminatory.

2. Evaluate conclusions

   1. Sleep and headaches conclusion: What is correct: the direction (negative, meaning more sleep associated with fewer headaches) is reasonable to state. What is flawed: “proves” is too strong — observational data cannot prove causation. “Cures” implies a medical treatment claim that requires controlled trials. A better conclusion: “The data suggests a moderate negative association between sleep and headache frequency in teenagers. Further controlled research is needed to investigate causation.”
   2. Coffee and productivity conclusion: This is a well-written conclusion. It is correct because it accurately describes the weak positive correlation, acknowledges the small sample as a limitation, and does not claim causation. It appropriately recommends further research.
   3. Chocolate and Nobel Prizes: This is a flawed conclusion. While the positive correlation may be real, it is almost certainly driven by confounding variables — wealthier, more developed nations both consume more chocolate AND invest more in science and education (which leads to Nobel Prizes). The conclusion incorrectly implies causation and gives impractical advice.

3. Problems with data collection

   1. Elite private school: Problem: biased sample. Elite private school students are not representative of all Queensland students — they likely have access to greater resources, smaller class sizes, and more support. The results cannot be generalised to all Queensland students.
   2. Gym survey: Problem: selection bias (voluntary response / self-selection). People at the gym are already active, so the data overrepresents fit people. The general population has many people who do not exercise, so conclusions about the general population would be misleading.
   3. Five best months: Problem: cherry-picking. Using only the best-performing months inflates the apparent positive relationship between advertising spend and sales. The true relationship may be weaker or only positive in certain conditions. This is ethically problematic if used to make business decisions.
   4. Social media poll: Problem: platform bias and self-selection. Posting the poll on social media means only people who use social media see it — those who use it heavily are more likely to respond. This over-represents heavy social media users and gives a biased picture of the relationship between social media use and academic performance.

4. Confounding variables

   1. Fire trucks and damage: Confounding variable: size of the fire. Larger fires cause more damage AND require more fire trucks. It is not that trucks cause damage — the fire’s severity drives both variables.
   2. Dentists and tooth decay: Confounding variable: national wealth. Wealthier countries can afford more dentists AND citizens have better diets and dental hygiene, leading to lower decay rates. Wealth drives both variables.
   3. Shoe size and reading ability (children): Confounding variable: age. Older children have bigger feet AND have had more years of schooling (so read better). Age is driving both shoe size and reading ability.
   4. Hospitals and death rate: Confounding variable: population size and health needs. Regions with larger or older populations need more hospitals AND have more deaths. The hospitals are not causing deaths — they are responding to a higher-need population.

5. Write statistical conclusions

   1. Year 9 homework vs maths score: Conclusion: The data from this class of 25 students suggests a moderate positive association between nightly homework hours and mathematics test scores. Students who completed more homework tended to score higher. Limitations: (1) The sample is a single class of 25 students, which is too small to generalise to all Year 9 students. (2) The presence of outliers (students with high homework but low scores, and low homework but high scores) suggests other factors influence test performance, such as prior knowledge, study quality, or test anxiety. (3) This is observational data — causation cannot be established. Recommendation: A larger study across multiple schools, controlling for prior ability, would give more reliable findings.
   2. Steps vs blood pressure: Conclusion: The data suggests a moderate negative association between daily step count and systolic blood pressure among adults aged 40–60, with a correlation coefficient of approximately −0.55. Adults who took more steps per day tended to have lower blood pressure. Limitations: (1) Participants were volunteers with fitness trackers, which may over-represent health-conscious individuals (self-selection bias). The results may not generalise to all adults. (2) Confounding variables such as diet, smoking, medication use, and stress levels were not controlled for. (3) Correlation does not imply causation — it is unclear whether walking reduces blood pressure or whether people with lower blood pressure are generally more able to walk. Recommendation: A controlled longitudinal study randomising participants into exercise programmes would better establish whether increased walking causes reductions in blood pressure.

6. Ethical and critical data analysis

   1. Misleading y-axis:
      1. Starting the y-axis at 90 compresses the visible range so that small differences in satisfaction scores appear much larger. A line that actually increases from 91 to 94 looks nearly vertical, exaggerating the apparent relationship between sales and satisfaction.
      2. Always check: the scale of both axes, whether the y-axis starts at zero, the range of data shown, whether only a selected time period or subset of data is displayed, and the source of the data.
   2. Cherry-picked employment data:
      1. The ethical problem is selective presentation: showing only the most favourable 3 months while hiding the broader 12-month picture. This manipulates the viewer’s perception and is deliberately misleading.
      2. Cherry-picking invalidates a statistical conclusion because it does not represent the full dataset. Any trend observed in a selected subset may be due to chance, seasonality, or short-term fluctuation rather than a genuine underlying relationship. Conclusions drawn from cherry-picked data cannot be trusted.
   3. Research plan for sleep vs reaction time: Data to collect: each student’s average hours of sleep the night before testing (independent variable) and their reaction time measured using a standardised computer-based test (dependent variable, e.g. milliseconds to click a button when a stimulus appears). Measurement: sleep could be self-reported via a brief survey; reaction time measured using the same device and test for all students. Participants: at least 80–100 Year 9 students from multiple Queensland schools to ensure a representative sample and sufficient power to detect a correlation. Avoiding bias: recruit students from different schools (urban/regional, public/private), test at the same time of day for all participants, ensure informed consent, and do not allow students to self-select — use random sampling or whole-class participation.

7. Interpreting r values

   1. r = 0.87 indicates a strong positive linear association. As one variable increases, the other tends to increase strongly. However, correlation does not imply causation — we cannot say one variable causes the other without a controlled experiment.
   2. Sleep vs doctor visits (r = −0.43):
      1. r = −0.43 indicates a moderate negative association: patients who slept more tended to visit the doctor less often.
      2. The headline overclaims. The r value shows a moderate association only — it does not establish causation. People with underlying health conditions may both sleep poorly AND visit the doctor frequently, making health status a confounding variable. The study is observational and cannot prove that sleep deprivation “sends” people to the doctor.
   3. Dataset B (150 data points) provides stronger evidence. A larger sample size makes the observed r value more stable and less likely to be due to random chance. With only 15 points (Dataset A), an r of 0.6 could easily arise by coincidence, whereas 150 points gives much more confidence that the association is real.
   4. No. r ≈ 0 only means there is no linear relationship. A clear non-linear relationship can still exist. Example: the relationship between x and y = x² is a perfect curve (U-shape), but r ≈ 0 because it is symmetric around x = 0. The variables are strongly related, just not in a straight-line fashion.

8. Comparing two studies

   1. Study B is more reliable because it has a much larger sample (1200 vs 25), uses random sampling from 40 schools (more representative of Queensland Year 10 students), and uses objective measurements (actual screen-time logs and a standardised test) rather than self-reports. A larger, more representative sample means the r value is less likely to be distorted by outliers or sampling bias.
   2. Study A uses self-reported screen time, which is subject to response bias — students may under-report their screen time. Study B avoids this by using a 2-week objective log. If students in Study A systematically under-report, the true range of screen time values is compressed, which could weaken or distort the measured r value.
   3. Concern 1: The decision is based on a single small study (25 students from one school) that cannot be generalised to all students. Concern 2: Correlation does not imply causation — even if high screen time is associated with lower reading, banning recess screens does not guarantee reading scores will improve. Other factors (home screen use, general engagement with reading, teacher quality) are not addressed by this policy.
   4. Likely confounding variable: socioeconomic status or parental involvement. Families with higher socioeconomic resources may both limit screen time AND invest more in books, tutoring, and reading activities. This would drive both lower screen time and higher reading scores, creating the negative correlation without screen time directly affecting reading ability.

9. Statistical report — running distance vs resting heart rate

   1. The data suggests a moderate negative association (r = −0.54) between weekly running distance and resting heart rate among recreational runners. Runners who covered more kilometres per week tended to have lower resting heart rates, though the relationship is moderate rather than strong, meaning running distance alone is not a perfect predictor of heart rate.
   2. Limitation 1: Self-selection bias — participants were recruited from a social media running group, meaning they are already active and engaged with running. This does not represent the general population and over-represents committed runners. Limitation 2: The social media recruitment means only people with internet access and interest in running groups responded, excluding casual or non-digital runners and non-runners entirely.
   3. Likely confounding variable: overall fitness level or general activity. People who are fitter overall both tend to run more AND have lower resting heart rates. It is not necessarily the running itself that lowers heart rate — overall cardiovascular fitness (influenced by diet, other exercise, genetics) could explain both.
   4. The advertising claim is statistically unjustified. The study shows only a moderate correlation (r = −0.54) from a self-selected sample, which cannot establish causation. The phrase “running makes your heart healthier” implies a causal, direct effect, which requires a controlled experiment (e.g. randomly assigning people to running and non-running groups). The current study is observational and the association may be driven by confounding variables.

10. Design and critique

    1. Study plan: Data to collect — count the number of books in each participant’s home (counted by the researcher or via a photo survey) and administer a standardised vocabulary test to each child. Sample size — at least 100–150 children from multiple schools across different socioeconomic areas to ensure representativeness. Sampling method — stratified random sampling across school year levels and suburbs. Minimising bias: (1) Use an objective book count rather than self-report to avoid under/over-counting. (2) Select schools from a range of socioeconomic areas to reduce confounding by family wealth.
    2. Three questions to ask: (1) Was the study observational or experimental? Observational data cannot establish causation. (2) What confounding variables could explain both x and y? A third variable may drive both trends. (3) What was the sample size and how were participants selected? A small or biased sample could produce a high r value by chance or due to sampling error.
    3. Pools and skin cancer:
       1. Most likely confounding variable: hours of sunshine / UV exposure (or wealth / coastal living).
       2. Suburbs with more sunshine and warmer climates tend to have more swimming pools (people want them) AND higher rates of skin cancer (due to UV exposure). The pools are not causing cancer — both variables are driven by the same underlying factor (sunny climate and outdoor lifestyle).
       3. A controlled experiment (randomised controlled trial) would be needed: randomly assign people to high and low UV exposure groups and measure skin cancer rates over time. In practice, ethical constraints prevent direct UV exposure experiments; instead, longitudinal cohort studies tracking known UV exposure levels alongside cancer outcomes provide the strongest available evidence.