Question at position 32 A regression of the amount of calories in a serving of breakfast cereal vs. the amount of fat gave the following results: Predicted Calories = 97.1053 + 9.6525(Fat) Which of the following is false?The correlation between amount of fat and calories is positive.If a cereal has 2 g of fat, then it is estimated that the total number of calories is about 116.If one cereal has 140 calories and 5 g of fat, then its residual is about 5 calories.It is estimated that in cereals with no fat, the total amount of calories is about 97.It is estimated that for every additional gram of fat in the cereal, the number of calories increases by about 10.

Question

BlackTom AI · Accepted Answer

Let’s break down what the regression equation tells us and then evaluate each statement against it.
The model is: Predicted Calories = 97.1053 + 9.6525 × Fat.
This means: (a) the intercept is about 97.1 calories when Fat = 0, (b) for each additional gram of fat, predicted calories increase by about 9.65 (roughly 10) calories, and (c) the sign of the relationship between fat and calories is positive since the slope is positive.
Now, evaluate each option one by one:
Option 1: The correlation between amount of fat and calories is positive.
Reasoning: The slope 9.6525 is positive, so as Fat increases, Predicted Calories increase. This indicates a positive relationship, hence this statement is true.
Option 2: If a cereal has 2 g of fat, then it is estimated that the total number of calories is about 116.
Reasoning: Predicted Calories = 97.1053 + 9.6525 × 2 ≈ 97.1053 + 19.305 ≈ 116.4103. Rounding to nearest whole calorie gives about 116. This statement is true.
Option 3: If one cereal has 140 calories and 5 g of fat, then its residual is about 5 calories.
Reasoning: Predicted Calories for Fat = 5 is 97.1053 + 9.6525 × 5 ≈ 97.1053 + 48.2625 ≈ 145.3678. Residual = Actual − Predicted = 140 − 145.3678 ≈ −5.3678, i.e., about −5 calories, not +5. Therefore this statement is false (the residual isn’t about +5; it’s about −5).
Option 4: It is estimated that in cereals with no fat, the total amount of calories is about 97.
Reasoning: With Fat = 0, Predicted Calories = 97.1053, which is about 97. This statement is true.
Option 5: It is estimated that for every additional gram of fat in the cereal, the number of calories increases by about 10.
Reasoning: The slope is 9.6525, which is approximately 10. Thus, this statement is true.
In summary, only Option 3 misstates the residual sign and magnitude; all other statements align with the regression output. The incorrect claim is that the residual would be about +5 calories when actual is 140 and fat is 5 g.

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