Predict in the right direction

Predict within the observed explanatory-variable range

Identify the explanatory variable and its observed range before substituting into a fitted model. Check whether the prediction is interpolation or extrapolation. Then explain why the sample and model are relevant to the situation being predicted.

Interpolation predicts a response for an explanatory value within the range used to fit the model. Extrapolation uses an explanatory value outside that range. The source’s twelve-point sketch distinguishes the middle interval from both outside intervals; see Interpolation and extrapolation.

The source calls interpolation “generally safe”, but being inside the range does not guarantee an accurate individual prediction. A suitable form of relationship, a relevant sample and appropriate measurement still matter. With only two observations, a straight line can pass through both without providing evidence that the underlying relationship is linear.

Explain the limits of extrapolation

Extrapolation is more uncertain because the observed relationship may change outside the sample range. The source warns against using an input of $400$ when the observed inputs run from $10$ to $60$. It also warns against applying one population’s fitted model to a different population.

Check the scatter and strength of linear association before using a straight-line prediction.

The source’s word “significant” is a modelling guideline here, not a reported statistical significance test.

Choose the regression direction from the modelling setup

If the problem fixes an explanatory variable, keep its fitted response model, even when solving it backwards for a calibration estimate. If neither variable has a prescribed explanatory role, regress the variable being predicted on the supplied predictor.

$x$ is prescribed explanatory

Estimate $y$ from $x$: Use $y$ on $x$

Estimate $x$ from $y$: Rearrange the same $y$-on-$x$ equation

$y$ is prescribed explanatory

Estimate $y$ from $x$: Rearrange the same $x$-on-$y$ equation

Estimate $x$ from $y$: Use $x$ on $y$

Neither is prescribed explanatory

Estimate $y$ from $x$: Use $y$ on $x$

Estimate $x$ from $y$: Use $x$ on $y$

These six completions fill the blank Student selection table, checked against the supplied worked directions in Example 7. Solving a fixed model backwards is not the same as fitting the other regression direction, and a zero fitted gradient cannot be inverted. Calibration also has its own uncertainty; the table is the source exercise’s selection convention, not a guarantee of optimal prediction in every statistical setting.