Question

match each correlation coefficient to the appropriate scatter plot.
correlation coefficients: r = 0.4, r = 0.9, r = -0.5, r = -1

Explanation:

Step1: Analyze Correlation Types

• Positive correlation: As \( x \) increases, \( y \) increases ( \( r>0 \) ). Strong positive: close to 1; weak positive: close to 0.
• Negative correlation: As \( x \) increases, \( y \) decreases ( \( r<0 \) ). Strong negative: close to -1; weak negative: close to 0.
• \( r = 1 \): Perfect positive linear relationship (all points on a line with positive slope).
• \( r = -1 \): Perfect negative linear relationship (all points on a line with negative slope).

Step2: Match Each Scatter Plot

1. Top - Left Plot: Points cluster closely along a vertical line? No, wait—wait, re - examine. Wait, maybe the first (top - left) has a strong positive? Wait, no, let's check the four plots:

• Top - Right Plot: Points show a positive trend, somewhat clustered. Maybe \( r = 0.9 \) (strong positive).
• Bottom - Left Plot: Points have a positive trend but more spread out? Maybe \( r = 0.4 \) (weak positive).
• Bottom - Right Plot: Points show a negative trend (as \( x \) increases, \( y \) decreases), somewhat clustered. Maybe \( r=- 0.5 \) (moderate negative).
• A plot with \( r = 1 \): All points on a straight line with positive slope. If there's a plot with points perfectly aligned in a positive line, that's \( r = 1 \). A plot with points perfectly aligned in a negative line is \( r=-1 \).

Assuming the plots:

• For \( r = 1 \): The plot where all points lie on a straight line with positive slope.
• For \( r=-1 \): The plot where all points lie on a straight line with negative slope.
• For \( r = 0.9 \): A scatter plot with a strong positive linear trend (points close to a line with positive slope).
• For \( r=-0.5 \): A scatter plot with a moderate negative linear trend (points show a negative trend, not too close to a line).
• For \( r = 0.4 \): A scatter plot with a weak positive linear trend (points have a positive trend but are more spread out).

Let's assume the plots:

• Plot with \( r = 1 \): Perfect positive line (identify the plot with all points on a positive - sloped line).
• Plot with \( r=-1 \): Perfect negative line (all points on a negative - sloped line).
• Plot with \( r = 0.9 \): Strong positive (points close to a positive line).
• Plot with \( r=-0.5 \): Moderate negative (points show negative trend, moderate spread).
• Plot with \( r = 0.4 \): Weak positive (points have positive trend, more spread).

Answer:

To match the correlation coefficients:

• \( r = 1 \): The scatter plot with all points on a straight line with a positive slope.
• \( r=-1 \): The scatter plot with all points on a straight line with a negative slope.
• \( r = 0.9 \): The scatter plot with a strong positive linear trend (points closely follow a positive - sloped line).
• \( r=-0.5 \): The scatter plot with a moderate negative linear trend (points show a negative trend, moderately clustered).
• \( r = 0.4 \): The scatter plot with a weak positive linear trend (points have a positive trend but are more spread out).

(Note: Since the exact plot details are a bit unclear from the image description, but the process is to use the definition of correlation coefficients and the pattern of scatter plots: positive/negative, strength (clustering around a line), and perfect linearity for \( r = \pm1 \).)