For a group of students in a statistics class, the scatter diagram compares their test scores ( $$ y $$ ) and the number of pencils they have ( $$ x $$. State whether there is no correlation or a negative correlation between the $$ x $$ and $$ y $$ variables. If there is a correlation, state whether it is most likely due to coincidence, a common underlying cause, or a direct cause. A. Negative correlation; direct cause B. Negative correlation; coincidence C. No correlation D. Negative correlation; common underlying cause

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1. The scatter diagram shows a negative relationship between the $$ x $$ variable (number of pencils) and the $$ y $$ variable (test scores). This means that as the number of pencils increases, the test scores tend to decrease.

2. However, it is important to consider whether this negative correlation implies a direct cause-and-effect relationship. In many cases where variables appear to be correlated, there might be a third factor influencing both. For example, an underlying factor such as the students’ organization or study habits might be influencing both the number of pencils they carry and their performance on tests.

3. Since having many pencils does not directly cause low test scores (nor do low test scores cause a student to carry more pencils), the negative correlation is most likely due to a common underlying cause rather than a direct cause.

Thus, the correct answer is:  
D. Negative correlation; common underlying cause.
There is a negative correlation between the number of pencils and test scores, and this correlation is most likely due to a common underlying cause.

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