Linear Regression and Residuals

Learn regression predictions, residuals.

Lesson 3 of 32 parts Practice Flashcards

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Regression Predictions

Objective: Explain regression predictions, apply the stated method, and check its conditions.

A fitted line ŷ=b₀+b₁x predicts the response from a specified explanatory input. Its coefficients are estimated from data, and individual observations can differ from predictions. Predictions outside the observed x range are extrapolations requiring additional justification. Check the observed input range before extrapolating.

Formulas

$ŷ=b₀+b₁x$

Worked examples

Try itA synthetic regression line is ŷ=6x+7. Find predicted y at x=7.Show answer

Step 1: Substitute x=7 into the fitted line. Step 2: Compute 6·7+7. Step 3: The prediction is 49; observed values can differ.

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Regression Predictions

Objective: Explain regression predictions, apply the stated method, and check its conditions.

Formulas

$ŷ=b₀+b₁x$

Worked examples

Try itA synthetic regression line is ŷ=6x+7. Find predicted y at x=7.Show answer

Step 1: Substitute x=7 into the fitted line. Step 2: Compute 6·7+7. Step 3: The prediction is 49; observed values can differ.

Residuals

Objective: Explain residuals, apply the stated method, and check its conditions.

A residual is observed response minus predicted response. Positive residuals lie above the fitted line and negative residuals below. Residual plots help diagnose curvature or changing spread; a small average residual alone does not establish model adequacy. Residual is observed minus predicted, not the reverse.

Formulas

$e=y−ŷ$

Worked examples

Try itA synthetic model predicts 49, while observed y is 56. Find the residual.Show answer

Step 1: Residual = observed minus predicted. Step 2: Compute (56)−(49). Step 3: The residual is 7, so the observation lies above the prediction.

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