Mathematics Standard • Year 12 • Module 5 • Lesson 5
Lines of Best Fit, Master
Mastery Challenge 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 nutrition study records servings of fruit x and a wellbeing score y: (1, 42), (2, 47), (3, 49), (4, 54), (5, 55), (6, 63). On graph paper, sketch these points and a line that passes through the middle of the scatter. Suggest an equation for your line and show two residuals to support your choice. 4 marks
2. Use the idea
Use your line from Q1 (or ŷ=38+4x) to predict the score at x=7 servings. The observed range is 1 to 6. Classify the prediction and state one caution. 2 marks
3. Check your reasoning
Two students draw different lines through the same scatterplot. State two criteria you would use to judge which is a better line of fit. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (4 marks)
One reasonable by-eye choice is ŷ=38+4x, which predicts 42, 46, 50, 54, 58, 62. Residuals y−ŷ are 0, +1, −1, 0, −3, +1; the line captures the positive trend, though the x=5 point sits below it. Other balanced lines may be acceptable if supported by residuals.
2. Use the idea (2 marks)
Using ŷ=38+4x gives ŷ=66. Since x=7 is outside 1 to 6, this is extrapolation; the pattern may not continue beyond the observed data.
3. Check your reasoning (2 marks)
A good line follows the overall direction/form and passes through the middle of the points, with residuals generally small and reasonably balanced above and below. It should not be chosen solely to pass through one unusual point.