Question

1. for which data set is the correlation coefficient ( r ) closest to 0?
2. for which data set is the correlation coefficient ( r ) closest to 1?
3. which data set indicates the strongest negative linear relationship between its two variables?

Explanation:

Step1: Analyze correlation coefficient meaning

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. \( r = 0 \) means no linear relationship, \( r = 1 \) means perfect positive linear relationship, and \( r=-1 \) means perfect negative linear relationship. Values close to 0 have weak linear relationships, close to 1 have strong positive, close to -1 have strong negative.

Step2: Analyze each figure for \( r \) closest to 0

• Figure 1: The points are scattered randomly, no clear linear trend (positive or negative).
• Figure 2: Points have some scatter but maybe a slight trend, but less random than Figure 1? Wait, no—Figure 1's points are more spread out with no linear pattern.
• Figure 3: Has a negative linear trend (as x increases, y decreases).
• Figure 4: Has a positive linear trend (as x increases, y increases).

So Figure 1 has the weakest linear relationship, so \( r \) closest to 0.

Step3: Analyze for \( r \) closest to 1

A perfect positive linear relationship has points close to a straight line with positive slope. Figure 4: As x increases, y increases, and points are close to a straight line (positive slope), more so than others. Figure 2 has more scatter. So Figure 4 has \( r \) closest to 1.

Step4: Analyze for strongest negative linear relationship

Strongest negative means points close to a straight line with negative slope. Figure 3: As x increases, y decreases, and points are close to a straight line (negative slope), more so than others. Figure 1 is random, Figure 2 has mixed, Figure 4 positive. So Figure 3 has strongest negative linear relationship.

Answer:

s:

1. For \( r \) closest to 0: Figure 1
2. For \( r \) closest to 1: Figure 4
3. For strongest negative linear relationship: Figure 3

(Note: If the options in the dropdowns are labeled as Figure 1, Figure 2, Figure 3, Figure 4, then select accordingly. The above is the analysis to determine each.)