Question

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 the model you can estimate a cereals 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?

in this context, what does it mean to say that a cereal has a negative residual?

a. the cereal is an outlier in the model.

b. the potassium content is actually lower than the model predicts for a cereal with that much fiber.

c. the cereal does not have enough potassium.

d. the potassium content is actually higher than the model predicts for a cereal with that much fiber.

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 less potassium than the model predicts.

b. a residual of -22 means that particular cereal has 22 g more fiber than the model expects.

c. a residual of -22 means that particular cereal has 22 mg more potassium than the model predicts.

d. a residual of -22 means that particular cereal has 22 g less fiber than the model expects.

Explanation:

• For the first question:
• The formula for the residual is \(Residual = Observed - Predicted\).
• If the residual is negative (\(Residual<0\)), then \(Observed - Predicted<0\), which means \(Observed < Predicted\).
• In the context of the regression model \(Potassium = 38+27Fiber\) (where the predicted value of potassium is based on the amount of fiber), a negative residual implies that the actual potassium content (observed value) is lower than the value predicted by the model for a given amount of fiber.
• For the second question:
• Given the residual formula \(Residual = Observed - Predicted\) and the residual is \(- 22\).
• Since the response variable in the model is potassium (in milligrams), \(Residual=-22\) means \(Observed - Predicted=-22\), or \(Observed=Predicted - 22\). So the particular cereal has \(22\) mg less potassium than the model predicts.

Answer:

• First question: B. The potassium content is actually lower than the model predicts for a cereal with that much fiber.
• Second question: A. A residual of - 22 means that particular cereal has 22 mg less potassium than the model predicts.