Module 6 · Regression & Correlation · Topic 8
Interpolation vs Extrapolation. Interpolation predicts within the range of observed x values and is generally reliable. Extrapolation predicts outside that range and is risky because the linear trend may not continue.
Prediction Intervals. A prediction interval gives a range for an individual future observation. It is wider than a confidence interval for the mean response because it includes both estimation uncertainty and individual variability.
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.Regression: ŷ = 40 + 5.2x (study hours to exam score). Observed x range: 1 to 8 hours. Predicting the score for x = 5: ŷ = 40 + 5.2(5) = 66 (interpolation, reliable). Predicting for x = 15: ŷ = 40 + 5.2(15) = 118 (extrapolation, unreliable).
Never extrapolate far beyond your data. The relationship may change outside the observed range. Always report the range of x values used when sharing predictions.
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