Questions: (a) A piano instructor examined student progress. He noticed that when students spend less time playing video games, their memorization ability tends to increase. What can he conclude? There is a correlation between amount of time spent playing video games and memorization ability. There may or may not be causation. Further studies would have to be done to determine this. (b) An environmental study was conducted. It reported that houses with more inhabitants do not necessarily produce more or less trash. What can be determined? There is no correlation between number of inhabitants and amount of trash produced. (c) Data was collected at a pool. It showed that being older does not imply that a swimmer will stay for a shorter or longer amount of time. What is likely true? There is no correlation between age and length of stay.

Solution Steps

For scenario (a), the statement indicates that there is a correlation between the amount of time spent playing video games and memorization ability. The conclusion drawn is that there may or may not be causation, which suggests that while a relationship exists, it does not imply that one variable directly affects the other. Thus, we can express this as:

\[ \text{Correlation: } r \neq 0 \quad \text{(indicating a relationship)} \] \[ \text{Causation: } \text{unknown} \]

In scenario (b), it is stated that there is no correlation between the number of inhabitants and the amount of trash produced. This means that changes in the number of inhabitants do not affect the amount of trash produced. We can express this as:

\[ \text{Correlation: } r = 0 \quad \text{(indicating no relationship)} \]

For scenario (c), the data shows that there is no correlation between age and length of stay. Similar to scenario (b), this indicates that age does not influence the length of stay at the pool. We can express this as:

\[ \text{Correlation: } r = 0 \quad \text{(indicating no relationship)} \]

Final Answer