Calculate residuals and use a residual plot to assess the appropriateness of a linear model.

What this dot point is asking

A high correlation coefficient suggests a linear model might fit, but it does not prove the relationship is straight. Residual analysis is the proper check. It looks at what the line misses, point by point.

Calculating residuals

For each data point, find the predicted value $\hat{y}$ by substituting its $x$ into the regression equation, then subtract that from the actual $y$. A point exactly on the line has a residual of zero.

The residual plot

A residual plot graphs each residual (vertical axis) against the explanatory variable (horizontal axis). It magnifies the leftover pattern that the eye cannot see in the original scatterplot.

When the residual plot shows a smile or frown shaped curve, the data bends and a straight line systematically over-predicts in some regions and under-predicts in others. That is a signal to consider a transformation or a non-linear model instead.

Why correlation alone is not enough

Two datasets can share the same correlation coefficient while one is genuinely linear and the other is strongly curved. The number $r$ measures only the strength of the linear part, so a curved dataset can still produce a moderately high $r$. The residual plot is what exposes the curve, which is why it is the deciding test for whether the line is suitable.

A funnel shape, where residuals spread out more at one end, signals that the spread of the response changes across the data. The line may still capture the average trend, but predictions become less reliable where the residuals fan out.

A complete answer calculates residuals as actual minus predicted, describes the residual plot as showing either random scatter or a systematic pattern, and states clearly whether the linear model is appropriate based on that pattern.