Question

1. a linear regression was performed and the information about the data using technology is given below: ( y_1 sim mx_1 + b )

statistics
( r^2 = 0.119 )
( r = 0.345 )
parameters
( m = 0.559379 )
( b = 3.942 )
what does the correlation coefficient tell you about the association of the data? choose the correct statement below.

a. the data shows a positive correlation because ( m = 0.559379 ).

b. the data shows a strong positive correlation because the ( r )-value is close to 1.

c. the data shows a weak positive correlation because the ( r )-value is close to 0.

d. the data shows a weak positive correlation because ( r^2 = 0.119 ).

Explanation:

1. Recall the concept of correlation coefficient \( r \): The sign of \( r \) indicates the direction (positive or negative) of the linear relationship, and the magnitude (how close \( |r| \) is to 0 or 1) indicates the strength. A positive \( r \) means positive correlation, a negative \( r \) means negative correlation. Values close to 0 mean weak correlation, values close to 1 or -1 mean strong correlation.
2. Analyze each option:

• Option A: The slope \( m \) is not the correlation coefficient. The correlation coefficient is \( r \), so this is incorrect.
• Option B: The \( r \)-value here is \( 0.345 \), which is not close to 1 (values close to 1 are typically above 0.8, for example). So it's not a strong positive correlation, this is incorrect.
• Option C: The \( r \)-value is \( 0.345 \), which is positive (so positive correlation) and close to 0 (since \( 0.345 \) is much closer to 0 than to 1), so it indicates a weak positive correlation. This matches the properties of \( r \).
• Option D: \( r^2 \) (coefficient of determination) is related to \( r \) (since \( r^2 = r\times r \)), but the question is about the correlation coefficient \( r \), not \( r^2 \). So this is incorrect.

Answer:

C. The data shows a weak positive correlation because the \( r \)-value is close to 0.