Mathematics Standard • Year 12 • Module 5 • Lesson 6
The Least Squares Regression Line, Build
Scaffolded Practice 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 the points (1,2), (2,3), (3,5), (4,4), calculate x̄, ȳ, Sxx=Σ(x−x̄)² and Sxy=Σ(x−x̄)(y−ȳ). Then find the least-squares gradient b=Sxy/Sxx and intercept a=ȳ−b x̄. 5 marks
2. Use the idea
In the model ŷ=1.5+0.8x, state which variable is predicted and interpret the gradient, using generic x and y units. 2 marks
3. Check your reasoning
Interpret the intercept 1.5 in the model. What should you check before giving it a real-world meaning? 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (5 marks)
x̄=2.5 and ȳ=3.5. Sxx=2.25+0.25+0.25+2.25=5. Sxy=2.25+0.25+0.75+0.75=4. Thus b=4/5=0.8 and a=3.5−0.8(2.5)=1.5. The regression line is ŷ=1.5+0.8x.
2. Use the idea (2 marks)
The model predicts y from x. For each 1-unit increase in x, the predicted y increases by 0.8 y-units on average.
3. Check your reasoning (2 marks)
At x=0, the model predicts y=1.5. Check whether x=0 is meaningful and within or near the observed data range; otherwise the intercept may have no useful contextual interpretation.