Module 6 · Regression & Correlation · Topic 4
Regression Line. Simple linear regression fits a straight line ŷ = a + bx that best describes the linear relationship between an explanatory variable x and a response variable y.
Interpretation. The slope b is the average change in y for a one-unit increase in x. The intercept a is the predicted y when x = 0 (which may or may not be meaningful in context).
ŷ = a + bx, where b = r × (sᵧ / sₓ) and a = ȳ − b × x̄
yhat = a + bx — Line that minimizes the sum of squared residuals.r = sum((x_i-xbar)(y_i-ybar)) / sqrt(sum((x_i-xbar)^2)sum((y_i-ybar)^2)) — Measures direction and strength of a linear association.Study hours (x) predicting exam score (y). If b = 5.2 and a = 40, then ŷ = 40 + 5.2x. Each additional hour of study is associated with a 5.2-point increase in predicted score.
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