Mathematics Standard • Year 12 • Module 5 • Lesson 7
Predictions Using the Regression Line, Master
Mastery Challenge for Predictions Using the Regression Line. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
For a sample of plants, the model is ŷ=6+2.8x where x is weeks after planting and y is height in cm; observed x ranges from 1 to 9. Calculate predictions at x=7 and x=12, and classify each. 3 marks
2. Use the idea
A plant at 7 weeks is 29 cm tall. Find its residual. A plant at 12 weeks is 44 cm tall. Why should the second residual be interpreted more cautiously? 3 marks
3. Check your reasoning
A student writes “the model proves every plant grows 2.8 cm each week”. Correct the statement. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
At 7 weeks: ŷ=6+2.8(7)=25.6 cm, interpolation. At 12 weeks: ŷ=6+2.8(12)=39.6 cm, extrapolation.
2. Use the idea (3 marks)
At 7 weeks, residual=29−25.6=+3.4 cm, so it is 3.4 cm taller than predicted. The 12-week value is outside the observed x-range, so the fitted prediction is an extrapolation and may be unreliable.
3. Check your reasoning (2 marks)
The regression gradient describes the predicted average change in height per week for the observed data pattern. Individual plants vary, and the model does not prove that every plant grows at exactly 2.8 cm per week.