Final answer:
To determine if A and B are independent based on their variation with C, we must verify if the product of their probabilities equals the probability of their intersection.
Step-by-step explanation:
If A varies as C, and B varies as C, then we are discussing a scenario where two variables are both directly proportional to a third variable. This implies that as C changes, both A and B change in the same proportion. To determine which of the provided statements is false, we need to inspect the interdependence and mutual exclusiveness of A and B and their relationship with C. One assumption is that if 'A and C do not have any numbers in common so P(A AND C) = 0', then A and C are mutually exclusive. However, this statement pertains to the probability of two events and not the variation between two variables like A and C. Therefore, this statement does not help us with the question at hand. On the other hand, another assumption is that 'If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise'. This can lead to the misconception that A and B cannot be independent, but it's not always the case.
Independence between two variables A and B in terms of variation implies that changes in one variable do not affect the other. To test this, we use the hint 'Are A and B independent? Hint—Is P(A AND B) = P(A)P(B)? If P(A AND B) = P(A)P(B), then A and B are independent. If A and B both vary from C, it does not necessarily mean they are dependent or independent of each other without additional information.
In the context of hypothesis testing with paired samples, statements AC are misleading since sample sizes can be small and it's not a question of two sample means but rather the difference between the paired samples measures that is often of interest. In summary, we cannot directly infer the relationship between A and B solely based on their relationships with C. Without additional information on how A and B are related to each other, we cannot confirm whether they are dependent or independent. Furthermore, probabilities and variations deal with different aspects of relationships among variables, which should not be confused.