Statistics / Unit 4: Two-Variable Data
4.4 Predictions and Residuals
Notes
Plug an $x$-value into the regression line to get a prediction $\hat{y}$. A residual measures how far the actual value is from the prediction.
- Residual $= y - \hat{y}$ (actual minus predicted).
- A positive residual means the point is above the line (the model underpredicted). A negative residual means it is below the line (overpredicted).
- The residuals of a least-squares line add to 0.
- A residual plot should show no pattern. A curved pattern means a linear model is not appropriate.
- Example: $\hat{y} = 25 + 4x$, and a student with $x = 5$ scored 40. Then $\hat{y} = 45$ and the residual is $40 - 45 = -5$.
Practice
The line is $\hat{y} = 10 + 2x$. A point has $x = 8$ and $y = 30$. Its residual is:
$\hat{y} = 10 + 2(8) = 26$, and $30 - 26 = 4$.
A negative residual means the actual value was:
Residual $= y - \hat{y}$, so a negative residual means $y < \hat{y}$.
A residual plot shows a clear U-shaped pattern. This suggests that:
Residuals from a good linear model should look random with no pattern.
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