Residuals: Signed Differences and Model Form — Worked Solutions

Residual=29−24=5 minutes. The observed trip is 5 minutes longer than predicted, so the model underpredicts this trip’s duration by 5 minutes.

Residual=20−24=−4 minutes. The observation is 4 minutes below the prediction, so the model overpredicts, or overestimates, duration by 4 minutes.

Against distance, the pair is (3,−2), with x in kilometers. Against predicted duration, the pair is (11,−2), with horizontal minutes. In both cases the vertical variable is residual in minutes; it is neither observed duration nor predicted duration.

A scatters on both sides of zero without a clear systematic curve, supporting linear form for the observed data. B is U-shaped, with positive residuals at the ends and negative in the middle; this systematic curvature suggests a linear model is not the most appropriate form. A better supports a line, though apparent randomness alone does not verify every model assumption or guarantee predictions.

Predictions for x=1,2,3,4,5,6 are 7,9,11,13,15,17 minutes. Observations are 10,9,10,12,15,20 minutes, so residuals are 3,0,−1,−1,0,3 minutes. Every residual uses the observation and prediction at the same distance.

At 1 kilometer, residual +3 means the route took 3 minutes longer than predicted: the model underpredicts duration by 3 minutes. At 3 kilometers, residual −1 means the route took 1 minute less than predicted: the model overpredicts by 1 minute.

Against distance plot (1,3),(2,0),(3,−1),(4,−1),(5,0),(6,3). Against predicted duration use (7,3),(9,0),(11,−1),(13,−1),(15,0),(17,3). The vertical variable is residual in minutes in either plot; horizontal units are kilometers in the first and minutes in the second.

The residuals have a U-shaped curved pattern: positive at distances 1 and 6, negative at 3 and 4, and zero at 2 and 5. The model underpredicts at the ends and overpredicts in the middle. This systematic curvature suggests a simple linear form is not the most appropriate model for these observations.

The original measurements show an approximately straight increasing trend: for example, observed time rises from 9.6 seconds at 1 page to 20.4 seconds at 6 pages and 32.3 seconds at 12 pages, with the other jobs close to the same straight path. The data exhibit an approximately linear trend, so a simple linear form is reasonable to investigate rather than fitting one to an obviously curved association. This initial plot does not establish every modeling assumption.

The residuals lie on both sides of zero, ranging from −0.5 to +0.5 seconds, without a clear systematic curve or trend across job size. For example, at 3 pages the residual is +0.5 seconds, at 7 pages it is −0.5, and near the upper end the 10/11/12-page residuals are +0.3/−0.3/+0.3 seconds. Their apparent pattern-free scatter, together with the approximately linear original-data trend, supports the linear form as appropriate for these observed jobs. It does not prove that all assumptions hold, guarantee each prediction, or establish a cause-and-effect mechanism. Equivalent descriptions of the apparent lack of a systematic pattern can earn credit.