Let's analyze the scatter plots and address each question based on the relationships they show:

1. For which data set is the correlation coefficient $r$ closest to 0?

   • A correlation coefficient close to 0 suggests no strong linear relationship between the variables. From the scatter plots, Figure 1 shows a weak or non-linear pattern, meaning that $r$ would be close to 0 for this data set.

2. For which data set is the correlation coefficient $r$ closest to -1?

   • A correlation coefficient of $r = -1$ indicates a strong negative linear relationship. Figure 2 shows a clear negative linear pattern, where as $x$ increases, $y$ consistently decreases. This suggests that the correlation coefficient for Figure 2 is closest to -1.

3. Which data set indicates the strongest positive linear relationship between its two variables?

   • A strong positive linear relationship is represented by a clear upward trend where as $x$ increases, $y$ also increases consistently. Figure 4 displays this behavior most strongly, indicating the strongest positive linear relationship.

To summarize:

1. Figure 1 has $r$ closest to 0.
2. Figure 2 has $r$ closest to -1.
3. Figure 4 shows the strongest positive linear relationship.

Would you like further details on any of these explanations? Here are five related questions to expand your understanding:

1. How is the correlation coefficient $r$ mathematically calculated?
2. Can a correlation coefficient of 0 imply a non-linear relationship?
3. How do outliers affect the correlation coefficient?
4. Is it possible for a strong correlation to exist without causation?
5. What are the limitations of using scatter plots to interpret data relationships?

Tip: A correlation coefficient $r$ close to 1 or -1 implies a stronger linear relationship, either positive or negative, while values closer to 0 indicate weaker or no linear relationship.