A negative residual, the model aimed too high

Question

A model predicts a store's afternoon foot traffic from the temperature (a downward relationship, since very hot days keep people home):

$\widehat{\text{visitors}} = 100 - 2 \cdot (\text{temperature in }^\circ\text{C}).$

At $10^\circ$ C the store actually recorded $74$ visitors.

What is the residual, and what does its sign mean?

Accepted Answer

Start with the prediction at 10^\circC: \widehat{	ext{visitors}} = 100 - 2 	imes 10 = 100 - 20 = 80. The residual is observed minus predicted: 	ext{residual} = 74 - 80 = -6. The residual is -6, and the negative sign means the actual count fell below the line: the model over-predicted, expecting 80 but seeing only 74. So 6 fewer visitors showed up than the line guessed. Note that the model having a negative slope does not change how residuals work. The slope's sign describes the trend; the residual's sign describes whether one particular point sits above or below that trend line. Takeaway. …

A negative residual, the model aimed too high

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