Questions: For many people, breakfast cereal is an important source of fiber in their diets. Cereals also contain potassium, a mineral shown to be associated with maintaining a healthy blood pressure. An analysis of the amount of fiber (in grams) and the potassium content (in milligrams) in one serving of 77 breakfast cereals produced the regression model Potassium = 38+27 Fiber. From this model you can estimate a cereal's potassium content from the amount of fiber it contains. In this context, what does it mean to say that a cereal has a negative residual? What does it mean to say that a cereal has a residual of -22 ? What does it mean to say that a cereal has a residual of -22 ? A. A residual of -22 means that particular cereal has 22 mg more potassium than the model predicts. B. A residual of -22 means that particular cereal has 22 mg less potassium than the model predicts. C. A residual of -22 means that particular cereal has 22 g more fiber than the model expects. D. A residual of -22 means that particular cereal has 22 g less fiber than the model expects.

Solution Steps

To solve this problem, we need to understand the concept of residuals in the context of a regression model. A residual is the difference between the observed value and the value predicted by the model. If a cereal has a residual of -22, it means the observed potassium content is 22 milligrams less than what the model predicts.

1. Identify the regression model: Potassium = 38 + 27 * Fiber.
2. Understand that a residual of -22 means the observed potassium content is 22 milligrams less than the predicted value.
3. Therefore, the correct interpretation is that the cereal has 22 mg less potassium than the model predicts.

The regression model given is: \[ \text{Potassium} = 38 + 27 \times \text{Fiber} \]

For a cereal with a fiber content of 5 grams: \[ \text{Predicted Potassium} = 38 + 27 \times 5 = 38 + 135 = 173 \, \text{mg} \]

A residual of -22 means the observed potassium content is 22 milligrams less than the predicted value: \[ \text{Observed Potassium} = 173 \, \text{mg} - 22 \, \text{mg} = 151 \, \text{mg} \]

A residual of -22 means that the observed potassium content is 22 milligrams less than the model predicts. Therefore, the correct interpretation is: \[ \boxed{\text{B}} \]

Final Answer