Let's analyze the relationship between the radius and the circumference of the objects measured to determine the strength of the correlation and the causation.
### Step 1: Calculate the Correlation Coefficient
The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, we calculate the correlation coefficient for the given pairs of radius and circumference.
The given pairs are:
- (3, 18.8)
- (4, 25.1)
- (6, 37.7)
- (9, 56.5)
The correlation coefficient calculated from these data points is approximately [tex]\(0.9999989948784027\)[/tex].
### Step 2: Determine the Strength of the Correlation
Correlation strength is typically classified as:
- Weak: [tex]\( |\text{correlation coefficient}| \leq 0.3 \)[/tex]
- Moderate: [tex]\( 0.3 < |\text{correlation coefficient}| \leq 0.7 \)[/tex]
- Strong: [tex]\( |\text{correlation coefficient}| > 0.7 \)[/tex]
Given our correlation coefficient of approximately [tex]\(0.9999989948784027\)[/tex], which is very close to 1, we see that it significantly exceeds 0.7. Therefore, the correlation is strong.
### Step 3: Determine Causation
Causation indicates that one variable directly affects the other. In situations where the correlation coefficient is positive and the relationship is known to be likely causal, we generally infer causation.
Given our correlation coefficient is positive and very close to 1, and considering we are dealing with the geometric properties of circles where the circumference [tex]\(C\)[/tex] is predicted by the formula [tex]\(C = 2\pi r\)[/tex], it is reasonable to infer that there is a direct relationship between radius and circumference. Therefore, it is likely causal.
### Conclusion
Combining the strength of the correlation and the likelihood of causation, the best description is:
- It is a strong positive correlation, and it is likely causal.
Therefore, the correct option is:
- It is a strong positive correlation, and it is likely causal.
