Question

the scientist graphed the two sets of data and found that a positive correlation exists in each set. which statement explains whether there is a relationship between the height of the tree and the amount of fruit produced.

• although fertilizer is in both data sets and is positively correlated, only a weak correlation can exist between tree height and fruit yield.
• although fertilizer is in both data sets and is positively correlated, it is impossible for any correlation to exist between tree height and fruit yield.
• since fertilizer is in both data sets and is positively correlated to both tree height and fruit yield, it is likely that tree height and fruit yield are negatively correlated.
• since fertilizer is in both data sets and is positively correlated to both tree height and fruit yield, it is like that tree height and fruit yield are also positively correlated.

Explanation:

To determine the relationship between tree height and fruit yield, we analyze the correlation logic. If a third variable (fertilizer) is positively correlated with both tree height and fruit yield, by the transitive nature of positive correlation (when a variable is a common positive correlate), tree height and fruit yield are likely to be positively correlated. Let's evaluate the options:

• The first option claims only a weak correlation, but there's no basis for that from the given info.
• The second option says no correlation is possible, which is incorrect as positive correlation via a common variable suggests a relationship.
• The third option says negative correlation, which contradicts the positive correlation of the common variable (fertilizer) with both.
• The fourth option correctly states that since fertilizer is positively correlated with both tree height and fruit yield, tree height and fruit yield are likely positively correlated.

Answer:

D. Since fertilizer is in both data sets and is positively correlated to both tree height and fruit yield, it is likely that tree height and fruit yield are also positively correlated.