The uploaded images contain four data sets (Figures 1–4) with scatter plots and accompanying tables. The goal is to determine the relationships between the pairs of variables in each dataset using the sample correlation coefficient $r$. Here's how the questions can be answered based on the provided information:

(a) For which data set is the sample correlation coefficient $r$ closest to 0?

• Analysis: When $r$ is closest to 0, there is little to no linear relationship between the variables. This would correspond to a scatter plot that shows no clear trend or pattern.
• Answer: The dataset in Figure 1 (variables $x$ and $y$) shows no apparent linear relationship, so $r$ is closest to 0.

(b) Which data set has an apparent negative, but not perfect, linear relationship between its two variables?

• Analysis: A negative $r$ indicates a downward trend. If the trend is not perfect, the points do not all lie exactly on a straight line.
• Answer: The dataset in Figure 3 (variables $w$ and $t$) exhibits a clear negative linear trend, but the points are scattered, so it is not perfect.

(c) Which data set indicates the strongest positive linear relationship between its two variables?

• Analysis: A strong positive linear relationship occurs when the points form an almost perfect upward line. $r$ would be close to $+1$.
• Answer: The dataset in Figure 4 (variables $m$ and $n$) shows a strong upward trend, indicating the strongest positive linear relationship.

(d) For which data set is the sample correlation coefficient $r$ equal to $-1$?

• Analysis: $r = -1$ corresponds to a perfect negative linear relationship, where all points lie exactly on a downward-sloping straight line.
• Answer: The dataset in Figure 2 (variables $u$ and $v$) shows a perfect negative linear relationship, so $r = -1$.

Summary of Answers:

• (a) Figure 1
• (b) Figure 3
• (c) Figure 4
• (d) Figure 2

Would you like additional details or explanations on how $r$ is calculated or interpreted?

Further Questions to Explore:

1. What does a correlation coefficient of exactly $+1$ represent?
2. How does the shape of the scatter plot influence the magnitude of $r$?
3. What is the relationship between $r$ and the slope of the line of best fit?
4. Can $r$ indicate a non-linear relationship between variables? Why or why not?
5. How do outliers affect the value of $r$?

Tip:

The correlation coefficient $r$ only measures the strength and direction of linear relationships. For non-linear patterns, other statistical methods, like calculating the coefficient of determination ($R^2$), may be more appropriate.