Before using a regression model, inspect the scatter diagram and identify the variables in the fitted equation. A large correlation coefficient supports linear association between those variables; it does not establish that every proposed model is suitable.
| Fitted relation | What to examine |
|---|---|
| \(y=a+bx\) | Scatter and correlation of \(x\) with \(y\). |
| \(\ln y=a+bx\) | Scatter and correlation of \(x\) with \(\ln y\), with \(y>0\). |
| \(y=a+\dfrac{b}{x}\) | Scatter and correlation of \(\dfrac{1}{x}\) with \(y\), with \(x\ne0\). |
Bridge the idea
The original x–y scatter is curved. A transformed scatter of x against ln y is nearly linear, with stronger linear association. This supports fitting ln y on x when that model is contextually reasonable. The original x–y correlation alone does not assess the transformed fit.
If \(\ln y=a+bx\), recover \(y=e^{a+bx}\) after prediction. Respect the domain and original units. When comparing models, explain the relevant scatter pattern and correlation rather than saying only “the larger r is better”; compare closeness to \(+1\) or \(-1\) through \(|r|\).
Exam wording: “The transformed scatter is approximately linear and the correlation between x and ln y is close to 1 in magnitude, supporting this fitted model.” Curvature or an influential outlier still deserves attention; fit is evidence to assess, not a guarantee.
Need help? Join our JC Math tuition classes.
Learn more