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Multiple Regression Introduction worksheets

Use a supplied two-predictor equation to identify variables, predict, calculate residuals and SSE, and write the least-squares objective.

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Use the supplied coefficients; do not refit unless asked. Interpret each slope while holding the other predictor fixed.

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Make every model comparison auditable

Finite data tables, printed transformed values, and supplied coefficients let students reproduce every residual, SSE, and interpretation on paper.

Learning objective

What this activity builds

Interpret and calculate with an introductory two-predictor model while holding the other included predictor fixed.

Worked example

What does b1 mean in yhat=b0+b1x1+b2x2?

It is the predicted change for a one-unit increase in x1 while x2 is held fixed.

This comparison isolates the model's x1 coefficient, but does not by itself prove causation.

Teaching notes

Make the practice count

  • Keep observed and predicted values separate.
  • Write one squared residual per row.
  • Avoid adding coefficient inference beyond this introductory scope.

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Math SheetNameScoreTeacherDateMultiple Regression IntroductionUse the supplied coefficients. Interpret one slope while holding the other predictor fixed; no coefficient inference is required.1.In yhat=5+1x1-1x2, interpret coefficient 1 when x1 is study hours, x2 is prior practice score, and y is predicted quiz points.Reference: A multiple-regression slope compares predictions while holding the other included predictor fixed. It is an association unless causal conditions are separately justified.2.Using yhat=6+2x1-1x2, calculate residuals and SSE for rows (x1,x2,y): (1, 0, 9), (2, 2, 7), (3, 4, 8), (4, 1, 14), (5, 3, 12).Reference: For each printed row use residual y-yhat. Then SSE=sum residual^2. Use the supplied coefficients; do not refit them.Show substitutions and residual signs; use only printed values.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateMultiple Regression IntroductionUse the supplied coefficients. Interpret one slope while holding the other predictor fixed; no coefficient inference is required.3.For rows (x1,x2,y): (1, 1, 5), (2, 3, 7), (3, 0, 14), (4, 2, 13), (5, 4, 15), (6, 1, 22), write the least-squares objective in b0,b1,b2 for yhat=b0+b1x1+b2x2. Do not solve it.Reference: Substitute each row into sum[y-(b0+b1 x1+b2 x2)]^2. Keep every one of the 4-6 printed rows.4.A model predicts weekly plant growth y (cm) from sunlight x1 (hours/day) and water x2 (liters/week): yhat=5+4x1-1x2. Identify the response and both predictors.Reference: The response is the outcome on the left side. Predictors appear on the right side and are used to form predictions.Show substitutions and residual signs; use only printed values.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateMultiple Regression IntroductionUse the supplied coefficients. Interpret one slope while holding the other predictor fixed; no coefficient inference is required.5.For yhat=4+1x1-2x2 and observed row (x1,x2,y)=(1,0,4), calculate the prediction and residual y-yhat.Reference: Substitute both predictors into the supplied equation. Residual=observed-predicted, so its sign matters.6.For yhat=5+2x1-2x2 and observed row (x1,x2,y)=(1,1,5), calculate the prediction and residual y-yhat.Reference: Substitute both predictors into the supplied equation. Residual=observed-predicted, so its sign matters.Show substitutions and residual signs; use only printed values.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateMultiple Regression IntroductionUse the supplied coefficients. Interpret one slope while holding the other predictor fixed; no coefficient inference is required.7.In yhat=6+3x1-2x2, interpret coefficient 3 when x1 is training sessions, x2 is baseline time, and y is predicted race time seconds.Reference: A multiple-regression slope compares predictions while holding the other included predictor fixed. It is an association unless causal conditions are separately justified.8.For rows (x1,x2,y): (1, 3, 1), (2, 0, 12), (3, 2, 13), (4, 4, 11), (5, 1, 22), write the least-squares objective in b0,b1,b2 for yhat=b0+b1x1+b2x2. Do not solve it.Reference: Substitute each row into sum[y-(b0+b1 x1+b2 x2)]^2. Keep every one of the 4-6 printed rows.Show substitutions and residual signs; use only printed values.math-sheet.comPage 4 of 4

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