Question

shown below are the scatter plots for four different data sets.
answer the questions that follow. the same response may be the correct answer for more than one question.
(images of four scatter plots: figure 1, figure 2, figure 3, figure 4)

1. for which data set is the correlation coefficient ( r ) closest to 0?

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1. for which data set is the correlation coefficient ( r ) closest to ( -1 )?

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1. which data set indicates the strongest positive linear relationship between its two variables?

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Explanation:

Step1: Recall Correlation Concepts

The correlation coefficient \( r \) measures linear association: \( r \approx -1 \) (strong negative), \( r \approx 1 \) (strong positive), \( r \approx 0 \) (little to no linear association).

Step2: Analyze Each Figure

• Figure 1: Negative linear trend (points slope down), so \( r \) negative but not extreme.
• Figure 2: Points are scattered with no clear linear trend (most random), so \( r \approx 0 \).
• Figure 3: Points have a weak or no clear linear trend but more scattered than Figure 2? Wait, no—Figure 2's points are more randomly distributed. Wait, recheck: Figure 2 has points with no clear slope. Figure 3 has some spread but maybe? Wait, no—Figure 2: points are scattered without a linear pattern. Figure 3: maybe some, but Figure 2 is more random. Wait, the question 1: closest to 0. So Figure 2? Wait, no—wait Figure 3? Wait, let's re-express:

1. Closest to 0: Data with least linear trend. Figure 2? Wait, Figure 3? Wait, looking at the plots:

• Figure 1: Negative linear (slope down, points close to line) → \( r \) close to -1? No, wait Figure 1: points are in a negative linear pattern, so \( r \) negative but maybe moderate. Figure 4: positive linear (slope up, points close to line) → \( r \) close to 1. Figure 2: points are scattered with no clear linear trend (more random) → \( r \approx 0 \). Figure 3: points have some spread but maybe a weak trend? No, Figure 2 is more random. Wait, maybe Figure 2? Wait, no—wait the options: the figures are Figure 1,2,3,4.

1. Closest to -1: Strong negative linear trend. Figure 1: points form a negative slope, close to a line → \( r \) close to -1.

1. Strongest positive linear: Figure 4: points form a positive slope, close to a line → \( r \) close to 1, strongest positive.

Step3: Answer Each Question

1. For \( r \approx 0 \): Figure 2 (most random, no linear trend). Wait, no—wait Figure 3? Wait, maybe I mixed up. Let's correct:

• Question 1: Data with \( r \approx 0 \): Figure 2 (points are scattered, no clear linear trend).
• Question 2: Data with \( r \approx -1 \): Figure 1 (negative linear trend, points close to line).
• Question 3: Data with strongest positive linear: Figure 4 (positive linear trend, points close to line).

Answer:

1. Figure 2
2. Figure 1
3. Figure 4