Practice Residuals in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The difference between an observed value and its predicted value from a regression model: \text{residual} = y - \hat{y} (observed minus predicted).
A residual is how much the model got wrong for a specific data point. Positive residual means the actual value was higher than predicted; negative means it was lower. If you plot all residuals, the pattern (or lack thereof) tells you whether the model is appropriate.
Example 1
easyGiven \hat{y} = 2 + 3x, and observed point (4, 15): calculate the residual and interpret whether the model over- or under-predicts.
Example 2
mediumFive observed and predicted values: (y, \hat{y}): (10,8), (15,14), (12,13), (20,19), (8,11). Calculate all residuals, verify they sum to 0, and compute \sum e_i^2.
Example 3
easyA regression model gives residuals: \{3, -2, 4, -5, 0\}. Are these residuals consistent with a valid LSRL? Calculate \sum e_i and \sum e_i^2.
Example 4
hardA residual plot shows residuals increasing in magnitude as \hat{y} increases (fan shape). What assumption is violated, and what does this mean for the validity of hypothesis tests in regression?