Mathematics Standard • Year 12 • Module 5 • Lesson 6
The Least Squares Regression Line, Master
Mastery Challenge 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
For a dataset, x̄=4, ȳ=18, Sxx=10 and Sxy=25. Calculate the least-squares gradient and intercept, and write the regression equation ŷ=a+bx. 3 marks
2. Use the idea
If x is the number of deliveries and y is delivery time in minutes, interpret the gradient and intercept of ŷ=8+2.5x. Observed x-values are 2 to 10. 3 marks
3. Check your reasoning
Use ŷ=8+2.5x to predict y at x=6. If the observed value is 25 minutes, find the residual and explain its sign. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
b=Sxy/Sxx=25/10=2.5. a=ȳ−b x̄=18−2.5(4)=8. Therefore ŷ=8+2.5x.
2. Use the idea (3 marks)
Each additional delivery is associated with a predicted increase of 2.5 minutes in delivery time. At zero deliveries the model predicts 8 minutes, but x=0 is outside the observed range and may not have a meaningful interpretation.
3. Check your reasoning (2 marks)
ŷ=8+2.5(6)=23 minutes. Residual=25−23=+2 minutes; the observed delivery time is 2 minutes longer than predicted.