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: (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.
Showing a random 20 of 50 problems.
Example 1
easyPredicted , observed . What is the residual?
Example 2
easyA residual is . What does that mean about the data point?
Example 3
mediumThe residuals of a fitted regression are . Confirm that this could be a valid LSRL fit.
Example 4
hardAn LSRL fit gives residuals with and . Estimate the residual standard error (use degrees of freedom).
Example 5
mediumIf a residual is at a point where the predicted value is , what was the observed value?
Example 6
easyWhat is the sum of all residuals from a least-squares regression line?
Example 7
easyA model predicts but the observed value is . Compute the residual.
Example 8
easyA residual plot shows points scattered randomly around with no pattern. Does this support the linear model?
Example 9
mediumThe standard deviation of the residuals (denoted ) is reported as . Roughly interpret this value.
Example 10
mediumA residual plot shows residuals fanning out as grows. What is this called and what does it suggest?
Example 11
easyA regression line predicts for . The actual observation is . Find the residual.
Example 12
mediumFor an LSRL with intercept and slope fit on , . Verify the line passes through .
Example 13
mediumObserved values have predictions . Compute all three residuals.
Example 14
mediumPredicted values: . Observed : . Compute the mean of the residuals.
Example 15
easyGiven and observed point , compute the residual.
Example 16
easyUsing , find the residual for the point .
Example 17
easyGiven and observed , find the residual.
Example 18
easyA positive residual means the observed value is above or below the prediction?
Example 19
hardTwo students fit different lines to the same data. Student A reports residuals summing to ; Student B reports residuals summing to . Whose line could be the LSRL?
Example 20
hardA residual plot shows a curved (parabolic) pattern. What kind of model might better fit the data?