Notice a Prediction Outside the Data Range

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A model was fitted to measurements at x=2 through x=10. Which proposed prediction is extrapolation? Candidate answers: (A) Predicting at x=14 (B) Predicting at x=2 (C) Predicting at x=3 (D) Predicting at x=8.
• Independent check inputs: A model was fitted to measurements at x=2 through x=11. Which proposed prediction is extrapolation? Candidate answers: (A) Predicting at x=15 (B) Predicting at x=2 (C) Predicting at x=3 (D) Predicting at x=9.

Learn the idea

Paired measurements describe how two variables vary together. A trend line summarizes association, predicted outputs can be compared with observations using residuals, and the observed input range limits how confidently that trend can be extended. Interpolation predicts within measured input bounds. Extrapolation predicts outside those bounds, where a fitted pattern may fail because the data did not constrain that part of the model.

Here is the method on the supplied teaching case. 14 lies outside the measured interval 2 through 10. Extending a trend outside its data range is extrapolation and requires more caution than interpolation.

An equation can produce an extrapolated value without supporting its reliability. Mark the observed interval and identify outside inputs explicitly; additional data or contextual evidence would be needed to strengthen an extended prediction.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A model was fitted to measurements at x=2 through x=17. Which proposed prediction is extrapolation? Candidate answers: (A) Predicting at x=15 (B) Predicting at x=2 (C) Predicting at x=21 (D) Predicting at x=3. Show the mathematical reason, not only the selected answer. Expected result: Predicting at x=21. 21 lies outside the measured interval 2 through 17. Extending a trend outside its data range is extrapolation and requires more caution than interpolation.

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