Question

jim has started a new exercise program. he has monthly checkups where his percentage of body fat is measured. jim records his body fat percentage and the number of months he has been on the exercise program. he collects data for 10 months and finds a linear model to give the relationship between the time spent exercising and his percentage of body fat. the equation of the line is \\(\hat{y} = 17 - 1.25x\\), where \\(\hat{y}\\) is his percentage of body fat and \\(x\\) is the time spent exercising (in months). the residual plot is shown. based on the residual plot, is the linear model appropriate? \\(\bigcirc\\) no, there is no clear pattern in the residual plot. \\(\bigcirc\\) yes, there is no clear pattern in the residual plot. \\(\bigcirc\\) no, there is a clear pattern in the residual plot, indicating that the linear model is not appropriate. \\(\bigcirc\\) yes, his percentage of body fat clearly decreases for the first four months and then increases after that.

Explanation:

To determine if a linear model is appropriate, we analyze the residual plot. A linear model is appropriate when residuals (differences between observed and predicted values) have no clear pattern (random scatter). If there's a clear pattern, the linear model is not appropriate. The key here is that a random residual plot (no pattern) supports the linear model, while a patterned residual plot does not. The correct reasoning is that if there's no clear pattern in the residual plot, the linear model is appropriate. Among the options, "Yes, there is no clear pattern in the residual plot" is correct because a lack of pattern in residuals indicates the linear model fits well. Other options are incorrect: a patterned plot would reject the linear model, and the last option misinterprets the residual plot's purpose (residuals, not the original data, determine model appropriateness).

Answer:

B. Yes, there is no clear pattern in the residual plot.