Mathematics Standard • Year 12 • Module 5 • Lesson 6
The Least Squares Regression Line, Apply
Contextual Problem Set for The Least Squares Regression Line. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
A calculator gives the least-squares model ŷ=24+6.5x, where x is weekly practice sessions and y is a skill score. State the response variable, interpret the gradient with units, and interpret the intercept cautiously. 3 marks
2. Use the idea
For the model in Q1, find the predicted score for x=4 sessions. A student with 4 sessions scores 52. Calculate the residual y−ŷ and interpret its sign. 2 marks
3. Check your reasoning
The least-squares method is used instead of choosing a line by eye. State what quantity it minimises and what the residual means. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
The response is skill score. The gradient predicts an average increase of 6.5 score points per additional weekly practice session. The intercept predicts 24 points at zero sessions; use it only if zero sessions is meaningful and supported by the data range.
2. Use the idea (2 marks)
ŷ=24+6.5(4)=50. The residual is 52−50=+2 points. The student scored 2 points above the model prediction.
3. Check your reasoning (2 marks)
It minimises the sum of the squared vertical residuals, Σ(y−ŷ)². A residual is observed y minus predicted ŷ for an individual x-value.