Simple linear regression
Objective. Fit and diagnose a linear model.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Simple linear regression models the relationship between a response variable y and a single predictor x using the equation ŷ = b₀ + b₁x, where the coefficients are chosen to minimise the sum of squared residuals Σ(yᵢ − ŷᵢ)² via the least squares method. The slope b₁ = Σ(xᵢ − x̄)(yᵢ − ȳ)/Σ(xᵢ − x̄)² measures the average change in y per unit increase in x, while the intercept b₀ = ȳ − b₁x̄ ensures the line passes through the data's centroid. Diagnosing the fit involves examining residuals eᵢ = yᵢ − ŷᵢ for patterns, since a good model shows residuals scattered randomly around zero rather than any systematic curve or trend. The coefficient of determination R² reports the proportion of variance in y explained by the linear model.
Worked example
A simple linear regression gives slope b₁ = 2 and intercept b₀ = 5. Find the predicted value ŷ when x = 4.
- 1Recall the regression equation ŷ = b₀ + b₁x.
- 2Substitute the given values: ŷ = 5 + 2(4).
- 3Compute the product: 2 × 4 = 8.
- 4Add to the intercept: ŷ = 5 + 8.
Answer: ŷ = 13
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Practice
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Start practisingCommon mistake. Students fit a line and assume a high R² proves the model is appropriate without checking a residual plot; a strong curved relationship can still yield high R² while clearly violating the linearity assumption.
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