Question

a nutritionist is curious about how the concentration of a vitamin supplement changes as a function of time (in hours) since a pill has been swallowed. the nutritionist measures the concentration for six hours after the pill was swallowed. he calculates the equation of the least - squares regression line as $hat{y}=0.0083 - 0.0012x$, where $hat{y}$ is the concentration and $x$ is the number of hours since the pill was swallowed. the graph shown is the residual plot for this model where the residuals are measured in parts per thousand. based on the residual plot, is the linear model appropriate? no, there is no clear pattern in the residual plot. yes, there is no clear pattern in the residual plot. no, there is a clear pattern in the residual plot, indicating that the linear model is not appropriate. yes, clearly the concentration of the vitamin decreases for the first three hours and then increases after that.

Explanation:

A linear model is appropriate when there is no clear pattern in the residual plot. If there is a clear pattern (such as a curve), it indicates that the linear model is not capturing the relationship well. In this case, we need to check the options based on the rule of residual - plot interpretation.

• Option 1: Incorrect. A lack of clear pattern would mean the linear model is appropriate, not the opposite.
• Option 2: Incorrect. The statement is correct (no clear pattern), but the conclusion (linear model is appropriate) is wrong.
• Option 3: Correct. If there is a clear pattern in the residual plot, the linear model is not appropriate.
• Option 4: Incorrect. The description about the concentration is not relevant to residual - plot interpretation for linear - model appropriateness.

Answer:

No, there is a clear pattern in the residual plot, indicating that the linear model is not appropriate.