Which of these best describes a relationship with a correlation coefficient of [tex]$r=.75$[/tex]?
A. strong positive
B. strong negative
C. weak positive
Answer (1)
A correlation coefficient ( r ) measures the strength and direction of a linear relationship.
r = 0.75 indicates a positive relationship because it's positive.
The relationship is strong because 0.75 is closer to 1 than to 0.
Therefore, r = 0.75 describes a \boxed{\text{strong positive}} relationship.
Explanation
Understanding Correlation Coefficients The correlation coefficient, denoted as r , is a measure of the strength and direction of a linear relationship between two variables. The value of r always falls between -1 and +1. A positive value indicates a positive correlation, meaning that as one variable increases, the other tends to increase as well. A negative value indicates a negative correlation, meaning that as one variable increases, the other tends to decrease. The absolute value of r indicates the strength of the correlation. Values close to +1 or -1 indicate a strong correlation, while values close to 0 indicate a weak or no correlation.
Analyzing the Given Correlation Coefficient In this case, we are given a correlation coefficient of r = 0.75 . Since the value is positive, we know that the relationship is positive. To determine the strength, we look at the magnitude of r . A value of 0.75 is closer to 1 than to 0. Generally, a correlation coefficient between 0.5 and 1 (or -0.5 and -1) is considered a moderate to strong correlation. Therefore, r = 0.75 indicates a strong positive correlation.
Conclusion Based on the analysis, the best description for a relationship with a correlation coefficient of r = 0.75 is a strong positive correlation.
Examples
Understanding correlation coefficients is crucial in many real-world scenarios. For instance, in healthcare, a strong positive correlation between exercise and cardiovascular health indicates that increased physical activity is associated with better heart health. In finance, a correlation coefficient can help assess the relationship between different investments, such as stocks and bonds. If two stocks have a strong positive correlation, they tend to move in the same direction, which can inform investment diversification strategies. Similarly, in marketing, a correlation coefficient might reveal the relationship between advertising spending and sales revenue, helping businesses optimize their marketing efforts.
