Using the Least Squares Line for Prediction

Making Predictions

Once you have the regression equation \(\hat{y} = a + bx\), substitute a value of \(x\) to predict \(y\):

Example: Equation: \(\widehat{\text{score}} = 31.5 + 9.2 \times \text{hours}\)

Predict score for 5 hours: \(\hat{y} = 31.5 + 9.2(5) = 77.5\) marks.

Interpolation vs Extrapolation

Why is Extrapolation Risky?

The linear relationship observed within the data range may not continue beyond it. The true relationship may:
- Level off (reach a maximum/minimum)
- Change direction
- Follow a curve

Example: Predicting exam scores for 20 hours of study using \(\hat{y} = 31.5 + 9.2x\) gives \(\hat{y} = 215.5\) — clearly impossible for a test out of 100. This shows the danger of extrapolation.

Other Limitations of the Least Squares Line

1. Only Valid for Linear Associations

• If the scatterplot shows a curved pattern, a straight line is inappropriate
• Always check the scatterplot and residual plot first

2. Correlation ≠ Causation

• A good fit does not mean \(x\) causes \(y\)
• Predictions are statistical estimates only

3. Outliers Affect Predictions

• The least squares line is sensitive to outliers, which can distort predictions

4. Strength of Association

• If \(|r|\) is weak (e.g. \(r = 0.3\)), predictions will be unreliable
• Only use the line for prediction if \(|r|\) is reasonably strong

Worked Example

Context: A study finds \(\widehat{\text{fuel used}} = 2.3 + 0.08 \times \text{distance}\) (litres, km), based on data for distances 10–200 km.

KEY TAKEAWAY: Interpolation (within data range) is generally reliable. Extrapolation (outside data range) is unreliable — always identify which type of prediction you are making.

EXAM TIP: VCAA often gives a prediction scenario and asks whether it is interpolation or extrapolation, and to comment on its reliability. Always state the data range and whether the x-value falls inside or outside it.

VCAA FOCUS: The word “limitations” in the KK specifically invites discussion of extrapolation, weak association, non-linearity, and causation. Cover all relevant limitations in your answer.