Mathematics Standard • Year 12 • Module 5 • Lesson 5
Lines of Best Fit, Build
Scaffolded Practice for Lines of Best Fit. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
The data are (1,12), (2,15), (3,17), (4,21), (5,23). A proposed line is y=8.6+3x. Complete a table of predicted y and residual y−ŷ for each x, then comment on whether the line is a reasonable visual fit. 4 marks
2. Use the idea
Using y=8.6+3x, predict y when x=6. State whether that prediction is interpolation or extrapolation if the observed x-values range from 1 to 5. 2 marks
3. Check your reasoning
Explain why a line of best fit should balance points above and below it rather than pass through every plotted point. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (4 marks)
Predictions are 11.6, 14.6, 17.6, 20.6, 23.6. Residuals y−ŷ are +0.4, +0.4, −0.6, +0.4, −0.6. The line is a reasonable fit: residuals are small and occur on both sides overall. A drawn line should pass through the middle of the scatter, not every point.
2. Use the idea (2 marks)
ŷ=8.6+3(6)=26.6. Since 6 lies outside the observed range 1 to 5, it is extrapolation and should be treated cautiously.
3. Check your reasoning (2 marks)
Real data usually scatter around a trend. A balanced line represents the centre of the pattern; forcing it through every point is generally impossible and would not describe the overall association.