Why predicting far outside the data can mislead

Question

An ice cream stand fit a straight line predicting daily sales from the day's high temperature. The data used to fit it covered temperatures from $60^\circ$ F to $90^\circ$ F, and the line looked excellent over that range. A manager now uses the same line to predict sales at $130^\circ$ F.

Explain why this prediction should not be trusted, even though the line fit the historical data well.

Accepted Answer

The problem is extrapolation: using the line far outside the range of temperatures it was built from. Three things go wrong at once. No data to back it up. The line's slope and intercept were chosen to fit points between 60^\circ and 90^\circ. At 130^\circ there is not a single observation supporting the line's behavior. You are trusting a pattern in a region where you have never checked it. The straight-line assumption probably breaks. A line assumes sales keep rising by the same amount for every degree, forever. Real behavior rarely stays linear so far out. Past some point, extreme heat lik…

Why predicting far outside the data can mislead

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