To determine which correlation coefficient indicates a weak negative correlation, we need to understand the range and meaning of correlation coefficients.
1. The correlation coefficient, denoted as \( r \), ranges from -1 to 1.
- A value of \( r = 1 \) indicates a perfect positive correlation.
- A value of \( r = -1 \) indicates a perfect negative correlation.
- A value of \( r = 0 \) indicates no correlation.
- Positive values of \( r \) indicate positive correlations, where as one variable increases, the other variable also increases.
- Negative values of \( r \) indicate negative correlations, where as one variable increases, the other variable decreases.
2. The strength of the correlation is categorized as follows:
- \( |r| \) close to 1 indicates a strong correlation.
- \( |r| \) around 0.5 indicates a moderate correlation.
- \( |r| \) close to 0 indicates a weak correlation.
Now we analyze the given options in the context of a negative correlation:
A. \( r = 0.5 \): This value indicates a moderate positive correlation, not a negative correlation.
B. \( r = -2.0 \): This value is not within the valid range of correlation coefficients \([-1, 1]\), so it is not a valid correlation coefficient.
C. \( r = -0.2 \): This value indicates a weak negative correlation, since the magnitude is close to 0 and the sign is negative.
D. \( r = -0.8 \): This value indicates a strong negative correlation, since the magnitude is close to 1 and the sign is negative.
Therefore, the correlation coefficient that indicates a weak negative correlation is:
[tex]\[ \boxed{r = -0.2} \][/tex]
