To determine the nature of the relationship between two quantitative variables based on the correlation coefficient [tex]\( r \)[/tex], let me explain what the correlation coefficient represents.
1. Correlation Coefficient ([tex]\( r \)[/tex]): Correlation [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between two quantitative variables. It ranges from -1 to 1.
- [tex]\( r = 1 \)[/tex]: Perfect positive linear relationship.
- [tex]\( r = -1 \)[/tex]: Perfect negative linear relationship.
- [tex]\( r = 0 \)[/tex]: No linear relationship.
2. Interpreting [tex]\( r \)[/tex]:
- A positive value of [tex]\( r \)[/tex] indicates a positive relationship, where an increase in one variable generally leads to an increase in the other variable.
- A negative value of [tex]\( r \)[/tex] indicates a negative relationship, where an increase in one variable generally leads to a decrease in the other variable.
- An [tex]\( r \)[/tex] value close to 0 indicates a weak relationship.
- When [tex]\( r = 0 \)[/tex], it suggests that there is no linear relationship between the variables.
Given that the correlation [tex]\( r \)[/tex] between the two quantitative variables is [tex]\( r = 0 \)[/tex]:
- This specific value indicates that there is no linear relationship between the variables. Thus, the variables do not move in any predictable way with respect to each other.
Therefore, based on the value of [tex]\( r \)[/tex] being exactly 0, we can conclude that:
There is no relationship between the two variables.
