5.6 Using Linear Regression to Make Predictions

Predicting with the LSRL

Once you have the least squares regression line (LSRL), ŷ = a + bx, plug in an x-value to predict ŷ. This works best for interpolation — predicting within the range of observed x-values. Predictions outside that range require extrapolation, which is riskier because the relationship may not hold there.

Extrapolation danger

Extrapolation means predicting y for x-values far outside the data's range. The linear pattern observed might not continue, so these predictions can be unreliable or meaningless. Always check whether your x-value falls within or near the range of the original data before trusting a prediction.

Residuals and prediction accuracy

A residual = observed y − predicted ŷ. Residuals show how far off a prediction is for actual data points. Small residuals and a high r² value suggest predictions are more trustworthy, while large scatter or a low r² means predictions carry more uncertainty.

Key terms

Interpolation

Using the regression line to predict y for x-values within the range of the observed data.

Extrapolation

Using the regression line to predict y for x-values outside the observed data range; can be unreliable.

Predicted value (ŷ)

The y-value calculated by plugging an x-value into the LSRL equation ŷ = a + bx.

Residual

Observed y minus predicted ŷ; measures prediction error for a specific data point.

Why check the x-range before predicting?

To avoid extrapolation — predictions outside the data's range may not reflect the true relationship between variables.

Try an AP-style question

What is the predicted score for a student who studies 4 hours?

1. 14
2. 26
3. 50
4. 15.5