To determine which correlation coefficient indicates a strong positive correlation, we need to understand the scale and interpretation of the correlation coefficient, [tex]\( r \)[/tex]. The correlation coefficient, [tex]\( r \)[/tex], ranges from -1 to +1 and measures the strength and direction of a linear relationship between two variables:
- [tex]\( r = 1 \)[/tex] indicates a perfect positive linear relationship.
- [tex]\( r = -1 \)[/tex] indicates a perfect negative linear relationship.
- [tex]\( r = 0 \)[/tex] indicates no linear relationship.
Positive values of [tex]\( r \)[/tex] indicate a positive correlation, where an increase in one variable is associated with an increase in the other. The closer [tex]\( r \)[/tex] is to +1, the stronger the positive correlation. Conversely, negative values of [tex]\( r \)[/tex] indicate a negative correlation, where an increase in one variable is associated with a decrease in the other. The closer [tex]\( r \)[/tex] is to -1, the stronger the negative correlation.
Given the options:
A. [tex]\( r = +0.9 \)[/tex]
B. [tex]\( r = -0.1 \)[/tex]
C. [tex]\( r = -0.09 \)[/tex]
D. [tex]\( r = +0.5 \)[/tex]
Let's analyze each option:
- Option A: [tex]\( r = +0.9 \)[/tex]. This value is very close to +1, indicating a very strong positive correlation.
- Option B: [tex]\( r = -0.1 \)[/tex]. This value is close to 0, representing a very weak negative correlation.
- Option C: [tex]\( r = -0.09 \)[/tex]. This value is also close to 0, indicating an extremely weak negative correlation.
- Option D: [tex]\( r = +0.5 \)[/tex]. This value indicates a moderate positive correlation, as it is halfway between 0 and +1.
From the given options, the value [tex]\( r = +0.9 \)[/tex] most strongly indicates a positive correlation since it is the closest to +1.
Therefore, the correct answer is:
A. [tex]\( r = +0.9 \)[/tex]
