Mathematics Standard • Year 12 • Module 5 • Lesson 5
Lines of Best Fit, Apply
Contextual Problem Set 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
A cyclist records distance from home (km) after t hours: (0,0), (1,12), (2,25), (3,35), (4,49). A by-eye model is d=12t. Calculate the predicted distances and residuals d−d̂, then assess the fit. 4 marks
2. Use the idea
Interpret the gradient 12 in d=12t with units. Predict the distance after 2.5 hours and state why the prediction is within the data range. 3 marks
3. Check your reasoning
In a different ride dataset, suppose distance rises almost linearly until 3 hours and then flattens. What feature of that pattern should you mention when judging a straight-line model? 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (4 marks)
Predicted distances are 0, 12, 24, 36, 48 km. Residuals are 0, 0, +1, −1, +1 km. These are small and balanced around zero, so d=12t is a reasonable by-eye fit for this sample.
2. Use the idea (3 marks)
The model predicts an increase of 12 km in distance from home for each additional hour. At 2.5 hours, d=12(2.5)=30 km. The observed time range is 0 to 4 hours, so 2.5 is interpolation.
3. Check your reasoning (2 marks)
Mention possible curvature or a change in rate. A straight line may be only an approximate summary and can give less reliable predictions where the pattern bends.