To determine which correlation coefficient indicates that there is no correlation between the variables, it's important to understand the meaning of the correlation coefficient, denoted as [tex]\( r \)[/tex]. The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from [tex]\(-1\)[/tex] to [tex]\(+1\)[/tex], where:
- [tex]\( r = +1 \)[/tex]: Perfect positive linear correlation.
- [tex]\( r = -1 \)[/tex]: Perfect negative linear correlation.
- [tex]\( r = 0 \)[/tex]: No linear correlation.
Now let's analyze each option:
A. [tex]\( r = \pm 1 \)[/tex]
- [tex]\( r = +1 \)[/tex] indicates a perfect positive correlation.
- [tex]\( r = -1 \)[/tex] indicates a perfect negative correlation.
- Therefore, this option indicates perfect correlation, not no correlation.
B. [tex]\( r = -0.5 \)[/tex]
- This indicates a moderate negative correlation between the variables.
C. [tex]\( r = 0.08 \)[/tex]
- This value is very close to 0. It indicates a very weak correlation, which is the closest to no correlation among the given options.
D. [tex]\( r = +0.5 \)[/tex]
- This indicates a moderate positive correlation between the variables.
Since [tex]\( r = 0.08 \)[/tex] is closest to zero, indicating very weak correlation and thereby approximating no correlation, the correct option is:
C. [tex]\( r = 0.08 \)[/tex]
