Mathematics Standard • Year 12 • Module 5 • Lesson 8
Bivariate Data Analysis Synthesis, Master
Mastery Challenge for Bivariate Data Analysis Synthesis. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
A council studies tree canopy x (%) and summer surface temperature y (°C). The sample has r=−0.78 and model ŷ=38−0.12x for canopy values from 10% to 60%. At x=40%, predict y and describe the association. 3 marks
2. Use the idea
Give one confounder that could affect canopy and temperature, and state what additional evidence would support a causal claim. 3 marks
3. Check your reasoning
Write a five-part analysis outline for this study: form, strength/direction, r, regression/prediction, and causation limitation. 3 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
ŷ=38−0.12(40)=33.2°C. The sample shows a moderately strong negative linear association: areas with more canopy tend to have lower summer surface temperatures. This is interpolation.
2. Use the idea (3 marks)
Building density or proximity to water could affect both canopy and temperature. Stronger causal evidence could come from a well-designed intervention or longitudinal comparison that accounts for these factors; correlation and regression alone do not establish causation.
3. Check your reasoning (3 marks)
(1) The pattern is approximately linear. (2) It is moderately strong and negative. (3) r=−0.78 quantifies a fairly strong negative linear association. (4) At 40% canopy, predicted temperature is 33.2°C. (5) This is an association, not proof of causation; confounders such as building density may contribute.