Scatter Plots and Correlation — Solutions

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1. Direction of correlation

   1. Temperature vs hot drinks: Negative correlation
   2. Study hours vs exam mark: Positive correlation
   3. Goals scored vs boot colour: No correlation
   4. Distance from city vs house price: Negative correlation
   5. Car age vs fuel efficiency: Negative correlation
   6. Age vs social media posts: No correlation
   7. Rainy days vs umbrella sales: Positive correlation
   8. TV hours vs fitness scores: Negative correlation

2. Strength and direction

   1. Tight cluster, upward: Strong positive correlation
   2. Downward trend, wide spread: Weak negative correlation
   3. No pattern: No correlation
   4. Generally upward, considerable scatter: Moderate positive correlation
   5. Points closely follow downward line: Strong negative correlation
   6. Faint upward tendency: Weak positive correlation

3. Identify independent and dependent variables

   1. Swimmer: Independent: training hours; Dependent: race time
   2. Ice-cream: Independent: daily temperature; Dependent: ice-cream sales
   3. Fertiliser: Independent: amount of fertiliser; Dependent: crop yield
   4. Attendance: Independent: attendance rate; Dependent: final grade
   5. Bridge: Independent: load (tonnes); Dependent: deflection amount
   6. Sugar intake: Independent: daily sugar intake; Dependent: resting heart rate

4. Causation vs correlation

   1. Swimming pools and income: Confuses correlation with causation. Wealthier suburbs can afford more pools AND have higher income — wealth is the confounding variable. A pool does not cause income to rise.
   2. Breakfast and test scores: May confuse correlation with causation. Students with good habits (nutrition, sleep) may perform better overall — general healthy habits could be a confounding variable. The claim of direct causation is too strong.
   3. Televisions and life expectancy: Confuses correlation with causation. Wealthier nations have more TVs AND better healthcare — national wealth is the confounding variable. TVs do not cause longer life.
   4. Ice cream and drowning: Classic example of correlation without causation. Hot weather drives both ice-cream sales and swimming (increasing drowning risk). Temperature is the confounding variable. Ice cream does not cause drowning.

5. Match descriptions to scatter plot types

   1. Shoe size vs height, clear upward trend: Strong positive correlation
   2. TV ads vs sleep time, vague downward drift: Weak negative correlation
   3. Birth month vs salary, evenly spread: No correlation
   4. Exercise vs resting heart rate, upward with moderate scatter: Moderate positive correlation (Note: in reality this might be negative for athletes, but the description states upward — accept moderate positive based on the described pattern)

6. Real-world data interpretation

   1. Libraries and crime rate:
      1. Strong negative correlation means: as the number of libraries increases, the crime rate tends to decrease, and this relationship is strong (points cluster closely around the trend).
      2. Possible confounding variable: socioeconomic status. Wealthier suburbs can invest in both more libraries AND have lower crime rates due to better employment and education opportunities.
      3. No, the journalist’s conclusion is not justified. Correlation does not imply causation. The data shows an association, not that libraries directly reduce crime. Other factors likely drive both variables.
   2. Screen time vs physical activity outlier:
      1. Yes, it is likely an outlier. Most teenagers with 8 hours of screen time would be expected to have low physical activity (following the negative trend), but this individual has 4 hours — much higher than expected — so the point falls far from the trend line.
      2. The outlier could weaken the apparent strength of the negative correlation, making it appear less strong or shifting the trend line upward on the right side.
      3. Possible reason: the teenager participates in a sport or structured physical activity programme that occupies several hours, independent of screen time habits.
   3. Fruit and blood pressure headline:
      1. Problem 1: The headline implies causation (“raises”) when the data only shows a weak positive correlation — no causal relationship has been established. Problem 2: The study involved only 30 patients, which is a very small sample — the headline generalises too broadly.
      2. Suggested headline: “Small study finds weak link between fruit intake and blood pressure — researchers call for further investigation.”

7. Design and interpret a bivariate study

   1. x-axis variable: Strength training hours per week should be on the x-axis. It is the independent variable — the one being deliberately varied (or selected) to see if it predicts maximum lift.
   2. Why 6 elite + 6 beginners is misleading: The two groups are very different in ability. The data will cluster into two separate groups (high training/high lift for elites, low training/low lift for beginners), creating an artificially strong positive correlation that reflects the difference between groups, not a continuous relationship within a single population.
   3. Careful conclusion: There is a strong positive correlation between weekly strength training hours and maximum lift capacity. However, we cannot conclude that more training causes greater lift. One possible confounding variable is natural strength or genetics — athletes with greater natural ability may train more and lift more.

8. Outliers and their effect on correlation

   1. Why (1, 5) is an outlier: With a strong negative correlation, a person with only 1 hour of social media use would be expected to have high sleep (8+ hours). The point (1, 5) shows only 5 hours of sleep despite low social media use — it lies far from the trend line even though both values individually seem plausible.
   2. Effect on correlation strength: The outlier would weaken the apparent strength of the negative correlation. It increases scatter around the trend line and may also pull the trend line slightly upward on the left side, making the correlation appear less strong or less negative than it truly is.
   3. Should the outlier be removed?: The researcher should not simply remove it. They should first investigate whether it is a data entry error. If it is genuine, it should be kept in the dataset and reported separately as an outlier. Removing real data without justification is misleading.

9. Interpret bivariate data from a table of values

   1. Describe correlation: Strong positive correlation. As traffic volume increases, PM2.5 concentration increases in a close, consistent pattern. Points would cluster tightly along an upward trend line.
   2. Independent and dependent variables: Independent (x-axis): traffic volume (thousands of vehicles). Dependent (y-axis): PM2.5 concentration (μg/m³). Traffic is the explanatory variable; air quality responds to it.
   3. Critique the planner’s claim: The claim is too strong. The data shows a positive correlation between traffic and PM2.5 levels, but correlation does not prove causation. Other factors — industrial activity, wind patterns, seasonal effects — could influence both variables. A controlled study would be needed to establish a causal link.

10. Compare and contrast two bivariate studies

    1. Which study is more reliable?: Study A (200 adults) is more reliable. A larger sample reduces the effect of random variation and makes it less likely that the observed correlation is due to chance. Study B’s 12 students is too small to draw strong conclusions.
    2. Can Study B conclude exercise causes lower BMI?: No. A strong negative correlation means BMI tends to be lower in people who exercise more, but it does not establish causation. Other factors (diet, genetics, baseline fitness, socioeconomic status) could explain both higher exercise and lower BMI.
    3. Confounding variables: Study A: chronic illness — older adults are more likely to have conditions requiring medical visits. Study B: diet — students who exercise more may also eat more healthily, which independently lowers BMI.