To determine which correlation coefficient indicates that there is no correlation between the variables, let's analyze each option provided:
A. [tex]\( r = \pm 1 \)[/tex]
- A correlation coefficient [tex]\( r = 1 \)[/tex] denotes a perfect positive correlation, meaning the variables move in exact unison. On the other hand, [tex]\( r = -1 \)[/tex] denotes a perfect negative correlation, meaning the variables move in exact opposition. Therefore, [tex]\( r = \pm 1 \)[/tex] indicates a very strong correlation, either positive or negative, but not no correlation.
B. [tex]\( r = +0.5 \)[/tex]
- A correlation coefficient [tex]\( r = +0.5 \)[/tex] denotes a moderate positive correlation, meaning there is some degree of association between the variables, but it is not perfect. Therefore, it indicates a moderate correlation, not no correlation.
C. [tex]\( r = 0.08 \)[/tex]
- A correlation coefficient [tex]\( r = 0.08 \)[/tex] denotes a very weak correlation. The value is very close to zero, implying that the linear relationship between the variables is extremely weak and can be considered almost negligible. Therefore, this correlation coefficient is indicative of no correlation, or a very weak correlation.
D. [tex]\( r = -0.5 \)[/tex]
- A correlation coefficient [tex]\( r = -0.5 \)[/tex] denotes a moderate negative correlation, implying that as one variable increases, the other variable tends to decrease at a moderate rate. Thus, it indicates a moderate correlation, not no correlation.
Given these analyses, the correlation coefficient indicating no correlation between the variables is:
C. [tex]\( r = 0.08 \)[/tex]
