Quick questions on Interpolation and extrapolation with a regression line for HSC Maths Standard 2

Interpolation is making a prediction at an $x$ value inside the range of the observed data. If the data covers $x = 2$ to $x = 20$, then any prediction with $x$ between $2$ and $20$ is interpolation.

Extrapolation is making a prediction at an $x$ value outside the range of the observed data. If the data covers $x = 2$ to $x = 20$, then a prediction at $x = 25$ or $x = 0$ is extrapolation.

1. State whether the prediction is interpolation or extrapolation. 2. Comment on whether the relationship is likely to continue (give a context-specific reason).

:::worked Worked example ### Classifying predictions

A linear regression of population for a regional NSW town from 1990 to 2020 might fit well over those $30$ years. Extrapolating to 2050 (interpolation? No, extrapolation): the population may continue to grow linearly, accelerate (if a major employer opens), stagnate (if drought or industry decline), or shrink. The line cannot capture any of these structural changes.

A regression of monthly cafe revenue against months since opening fits well for the first $24$ months. Extrapolating to month $60$: the linear model assumes growth continues at the same rate forever. In practice, the cafe will eventually saturate the local market. The model is unreliable for that horizon.

Interpolation gives the best linear prediction; it is not a guarantee. Some scatter around the line is expected.

Always state the $x$ range explicitly: "the data covers $x = 2$ to $x = 20$, so $x = 25$ is extrapolation".

Mild extrapolation can be reasonable; substantial extrapolation is the problem.

"Growth might change" is not a strong answer. Cite a specific reason: market saturation, biological limits, economic shifts. :::