Question

three scatterplots are shown, along with their best-fit lines and correlation coefficients, ( r_1 ), ( r_2 ), and ( r_3 ). choose the correct statement.
a ( r_3 < r_2 < r_1 )
b ( r_2 < r_3 < r_1 )
c ( r_3 < r_2 < r_1 )
d ( r_1 < 0 < r_2 )
e ( r_2 < r_3 < r_1 )

Explanation:

Step1: Recall Correlation Coefficient Concept

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. A value closer to \( 1 \) (or \( -1 \)) indicates a stronger linear relationship, and positive \( r \) means positive linear association (as \( x \) increases, \( y \) tends to increase).

Step2: Analyze Scatter Plots

• For \( r_1 \): The points are closest to the line, so \( r_1 \) is closest to \( 1 \).
• For \( r_2 \): Points are closer to the line than \( r_3 \) but less than \( r_1 \), so \( r_2 \) is between \( r_3 \) and \( r_1 \).
• For \( r_3 \): Points are most spread out from the line, so \( r_3 \) is the smallest among the three (but still positive since the line has positive slope).

So the order from smallest to largest is \( r_3 < r_2 < r_1 \)? Wait, no—wait, let's re - check. Wait, the first plot (with \( r_1 \)) has points very close to the line, so \( r_1 \) is near 1. The second plot (\( r_2 \)) has points closer than the third (\( r_3 \)). So the closer the points are to the line, the larger \( r \) (since positive correlation). So \( r_3 \) (most spread) \( < r_2 < r_1 \) (least spread, closest to line). Wait, but looking at the options: Option C is \( r_3 < r_2 < r_1 \). Wait, let's check the options again.

Wait the options:

A. \( r_1 < r_2 < r_3 \)

B. \( r_2 < r_3 < r_1 \)

C. \( r_3 < r_2 < r_1 \)

D. \( r_1 < 0 < r_2 \) (but all lines have positive slope, so \( r \) should be positive, so D is wrong)

E. \( r_2 < r_1 < r_3 \)

Since all three have positive correlation (lines have positive slope), so \( r_1, r_2, r_3 \) are positive. The strength: the closer the points to the line, the larger \( r \). So the first plot ( \( r_1 \)) has points closest to the line, so \( r_1 \) is largest. The third plot ( \( r_3 \)) has points most spread out, so \( r_3 \) is smallest. The second plot ( \( r_2 \)) is in between. So \( r_3 < r_2 < r_1 \), which is option C.

Answer:

C. \( r_3 < r_2 < r_1 \)