Alright, let's analyze the relationship between the number of park rangers and the number of forest fires using the given data.
### 1. Correlation Coefficient:
First, we need to understand what the correlation coefficient tells us. The correlation coefficient, which ranges from -1 to 1, measures the strength and direction of a linear relationship between two variables. Here:
- A value of 1 indicates a perfect positive linear relationship.
- A value of -1 indicates a perfect negative linear relationship.
- A value of 0 indicates no linear relationship.
Given the correlation coefficient:
[tex]\[ -0.5974 \][/tex]
- The negative sign indicates that there is an inverse relationship between the number of park rangers and the number of forest fires.
- The magnitude [tex]\(0.5974\)[/tex] suggests a moderate negative linear relationship.
### Real-World Interpretation:
In the context of park rangers and forest fires:
- As the number of park rangers increases, the number of forest fires tends to decrease, and vice versa.
- However, this relationship is not extremely strong, indicating that while there is a noticeable trend, other factors may also significantly impact the number of forest fires.
### 2. Mean Residual:
The mean residual provides another important aspect:
Given the mean residual:
[tex]\[ -0.00043 \][/tex]
- Residuals represent the difference between the observed values and the values predicted by the linear model.
- A mean residual close to 0 indicates that, on average, the model's predicted values are very close to the actual observed values.
### Real-World Interpretation:
- The model predicting forest fires based on the number of park rangers is quite accurate, considering the mean residual is nearly zero.
- However, individual residuals (both positive and negative) suggest that there are discrepancies for some data points, implying that factors other than the number of park rangers might be influential in those cases.
### Summary:
- There is a moderate negative correlation between the number of park rangers and the number of forest fires, suggesting that having more park rangers tends to be associated with fewer forest fires.
- The predictive model is fairly accurate on average (since the mean residual is very close to zero), but some predictions might still have significant deviations due to other contributing factors.
