- The correlation coefficient \(r = -0.78\) indicates a negative linear relationship between age and value, meaning older cars generally have lower values.
- The magnitude \(|r| = 0.78\) represents a strong linear association because it is greater than \(0.7\).
- The statement that correctly describes this is: \(\boxed{\text{There is a strong negative linear association: older cars tend to have lower resale values in this sample.}}\)
Explanation
- Understanding the correlation coefficient \(r\)
The correlation coefficient, denoted as \(r\), is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. In this scenario, we are given \(r = -0.78\) for the relationship between a car's age and its resale value, based on a sample of \(60\) used cars. To describe this relationship accurately, we need to interpret both the sign and the magnitude of \(r\).
- Interpreting the direction of the relationship
The sign of the correlation coefficient indicates the direction of the linear relationship.
- A positive value (\(r > 0\)) suggests that as one variable increases, the other variable also tends to increase.
- A negative value (\(r < 0\)) suggests that as one variable increases, the other variable tends to decrease.
Since our coefficient is \(r = -0.78\), which is negative, there is a negative linear association between a car's age and its resale value. This means that, generally, as a car gets older, its resale value decreases. The suggestion that a negative value implies a mistake in data entry is incorrect; a negative correlation is a meaningful statistical finding.
- Interpreting the strength of the relationship
The strength of the linear association is determined by the absolute value of the correlation coefficient, denoted as \(|r|\).
- \(|r| < 0.3\): weak association.
- \(0.3 \le |r| < 0.7\): moderate association.
- \(|r| \ge 0.7\): strong association.
For our case, \(|r| = |-0.78| = 0.78\). Since \(0.78 \ge 0.7\), we characterize the relationship as having a strong linear association.
- Concluding the best description of the correlation
Combining our findings, we have a negative correlation (\(r < 0\)) and a strong association (\(|r| \ge 0.7\)). Therefore, there is a strong negative linear association between a car's age and its resale value. Among the given options, the statement "There is a strong negative linear association: older cars tend to have lower resale values in this sample" best describes the relationship.
Examples
When analyzing data, such as the relationship between study hours and exam scores or the price of a house versus its square footage, understanding the correlation coefficient helps you interpret the underlying trends. In this case, the negative correlation between a car's age and its value allows potential buyers and sellers to predict that, on average, a vehicle will lose market value as it ages. Understanding this helps in making informed financial decisions.