Mathematics Standard • Year 12 • Module 5 • Lesson 8
Bivariate Data Analysis Synthesis, Apply
Contextual Problem Set 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 dataset compares age of a used car x (years) with sale price y ($1000s). It has r=−0.91 and fitted line ŷ=32−2.4x, based on ages 1 to 10 years. Describe the relationship and predict the price for an 8-year-old car. 3 marks
2. Use the idea
Interpret the gradient and state one limitation of the predicted price. Explain why this relationship alone does not identify a causal effect of age. 3 marks
3. Check your reasoning
A student uses the line to predict price at age 15. State the calculation and assess the prediction. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
There is a strong negative linear association: older cars in the sample tend to sell for less. ŷ=32−2.4(8)=12.8, so the predicted price is $12,800. This is interpolation.
2. Use the idea (3 marks)
The model predicts a decrease of $2,400 in sale price per additional year of age on average. Features, condition, kilometres and location also affect price; age is associated with these factors, so the model does not isolate a causal effect.
3. Check your reasoning (2 marks)
ŷ=32−2.4(15)=−4 ($1000s), or −$4,000. This is extrapolation beyond ages 1–10 and gives an impossible price, so the model should not be used there.