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 1?

select

1. for which data set does the correlation coefficient r appear to be equal to -1?

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1. which data set shows the least evidence of a linear relationship?

select

Explanation:

Question 1: For which data set is the correlation coefficient \( r \) closest to 1?

Step 1: Recall Correlation Coefficient Meaning

The correlation coefficient \( r \) measures the strength and direction of a linear relationship. \( r = 1 \) means a perfect positive linear relationship (points lie on an upward - sloping line).

Step 2: Analyze Each Figure

• Figure 1: The points show a negative linear trend (as \( x \) increases, \( y \) decreases), so \( r \) is negative, not close to 1.
• Figure 2: The points seem to follow an upward - sloping linear pattern closely. As \( x \) increases, \( y \) increases in a relatively linear way, which is consistent with a positive correlation close to 1.
• Figure 3: The points are scattered in a way that does not show a strong linear trend, and the pattern is not a clear positive linear one.
• Figure 4: The points show a negative trend (as \( x \) increases, \( y \) decreases on average), so \( r \) is negative, not close to 1.

Step 1: Recall \( r=-1 \) Meaning

\( r = - 1 \) means a perfect negative linear relationship (points lie on a downward - sloping line).

Step 2: Analyze Each Figure

• Figure 1: The points lie close to a straight, downward - sloping line. As \( x \) increases, \( y \) decreases in a very linear fashion, which is consistent with \( r=-1 \).
• Figure 2: Positive trend, so \( r \) is positive, not - 1.
• Figure 3: No clear linear trend, let alone a perfect negative one.
• Figure 4: The points have a negative trend but are not as close to a straight line as Figure 1.

Step 1: Recall Linear Relationship Evidence

A linear relationship is evidenced by points that cluster around a straight line (either positive or negative slope). The more scattered the points are from a straight - line pattern, the less evidence of a linear relationship.

Step 2: Analyze Each Figure

• Figure 1: Clear negative linear trend.
• Figure 2: Clear positive linear trend.
• Figure 3: The points are scattered in a way that does not follow a clear linear pattern (neither positive nor negative linear trend is strong).
• Figure 4: Has a negative linear trend (points follow a somewhat linear, downward - sloping pattern).

Answer:

Figure 2

Question 2: For which data set does the correlation coefficient \( r \) appear to be equal to - 1?