Topic Review — Bivariate Data — Solutions

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1. Scatter plots and correlation fluency

   1. Strong positive correlation. Taller people cast longer shadows — the relationship is very consistent.
   2. Independent variable: daily temperature (x-axis). Dependent variable: hot chocolate sales (y-axis).
   3. Strong negative correlation (tight cluster, downward slope from left to right).
   4. Example: a person’s shoe size and the number of times they laugh per day (many valid answers possible).
   5. An outlier is a point that lies far from the general pattern. It can make a strong correlation appear weaker, or shift the line of best fit away from the main data cluster.
   6. Likely confounding variable: age. Older people are often taller (up to adulthood) and tend to have more experience and seniority, which may drive higher earnings. Height itself likely does not cause higher income.

2. Line of best fit fluency

   1. y = 4(7) − 3 = 28 − 3 = 25
   2. y = −2.5(16) + 60 = −40 + 60 = 20
   3. Extrapolation. x = 30 is beyond the data range of 5–25.
   4. Gradient m = (42 − 12) ÷ (6 − 0) = 30 ÷ 6 = 5. y-intercept = 12. Equation: y = 5x + 12.
   5. False. A steeper gradient means y changes more rapidly with x, but it does not indicate how tightly clustered the points are around the line. Correlation strength is about how closely points follow the line, not how steep the line is.
   6. A negative gradient means the two variables are negatively correlated — as x increases, y tends to decrease.

3. Using the line of best fit

   1. Revision vs exam score (y = 3.2x + 28):
      1. y = 3.2(10) + 28 = 32 + 28 = 60 out of 100.
      2. Not very reliable. x = 25 hours is well beyond the data range (1–15 hours), making this extrapolation. The linear pattern may not continue at very high revision hours — there may be diminishing returns.
      3. For every additional hour of revision, the predicted exam score increases by 3.2 marks on average.
      4. The y-intercept of 28 represents the predicted score for a student who does zero hours of revision — essentially a base score reflecting prior knowledge or test-taking ability without specific study.

4. Interpretation and limitations

   1. Hand size and vocabulary: Problem 1: Sample size of 12 is very small — results are unreliable for broader conclusions. Problem 2: Likely confounding variable is age — older people tend to have larger hands AND larger vocabularies. The observed correlation may not be a direct relationship.
   2. Ice cream and sunburn: Ice cream does not cause sunburn. Both increase on hot, sunny days. The confounding variable is weather (hot, sunny conditions): warm weather drives more people to buy ice cream AND increases sun exposure, leading to more sunburn. The correlation is real but the causal claim is false.
   3. Caffeine and concentration survey bias: Self-selection bias: students who volunteer to complete the survey may not represent the wider Year 9 population. Additionally, self-rated concentration is subjective and prone to social desirability bias (students may report what they think sounds good).
   4. Housing prices extrapolation: This is extrapolation — predicting beyond the 6-month data range into the future. It is unreliable because housing markets are affected by many unpredictable factors (interest rates, economic conditions, policy changes) that may cause the trend to change direction or level off.

5. Extended problem solving

   1. Sprint time (y = −0.4x + 14.2):
      1. y = −0.4(7) + 14.2 = −2.8 + 14.2 = 11.4 seconds.
      2. y = −0.4(15) + 14.2 = −6 + 14.2 = 8.2 seconds. This is extrapolation (15 > 10, the upper data limit) and is less reliable. Training 15 sessions per week may cause fatigue or injury, breaking the linear pattern.
      3. For each additional training session per week, the predicted 100 m sprint time decreases by 0.4 seconds on average.
      4. Conclusion: The data from 35 junior athletes suggests a moderate negative association between training sessions per week and 100 m sprint time. Athletes who trained more frequently tended to run faster. Limitation: The study cannot establish causation — naturally faster athletes may also train more. Additionally, the study did not control for the quality or intensity of training sessions.
   2. Researcher A vs Researcher B:
      1. Researcher A’s results are more reliable. Reasons: (1) Much larger sample (400 vs 12) — larger samples give more stable estimates of correlation. (2) Multiple schools across Queensland — reduces bias from a single school’s characteristics and improves representativeness.
      2. With only 12 data points, a few extreme values or random variation can dramatically affect the calculated correlation. Researcher B’s small sample is much more susceptible to chance variation, making it appear stronger than the true population correlation. Researcher A’s large sample “averages out” this noise and gives a more accurate picture of the true strength.
   3. Student’s reasoning about LOBF: The reasoning is flawed. A line that passes through many data points is not necessarily the best fit — it may be overfitting or zig-zagging through the data. A good line of best fit: (1) has approximately equal numbers of points above and below it, (2) is positioned through the centre of the data cloud, (3) minimises the overall distance of points from the line, and (4) follows the general trend without being pulled toward individual points or outliers. Passing through many points can actually indicate the line is too specific to the sample.