Question

a forest management agency collects measurements from beech trees in a public park. the age of many of the trees is known, and the basal area - the area of land occupied by the trunks of the trees - is also measured. a least - squares equation that compares the basal area to the age of each tree is given by (widehat{\text{basal}}=0.439(\text{age}) + 6.226). based on the residual plot shown, is a linear model appropriate for comparing tree age to basal area? because the residuals are centered about zero, a linear model is appropriate. because the residual plot does not show a clear pattern, a linear model is appropriate. because the residual plot shows a random a scatter of points, a linear model is not appropriate. because the residual plot does not appear linear, it is likely that tree age and basal area do not have a linear relationship.

Explanation:

When evaluating if a linear model is appropriate, we look at the residual plot. A key criterion is the absence of a clear pattern. If residuals are randomly scattered (no curvature, no increasing/decreasing trend), a linear model is appropriate. Centering about zero (\(y = 0\)) is a good sign (no bias in prediction direction), but the main check is pattern - absence. A random scatter of points (no non - linear pattern) supports a linear model. If the residual plot had a curve (e.g., U - shape, inverted U - shape), a non - linear model would be better.

Answer:

Because the residual plot does not show a clear pattern, a linear model is appropriate.