Answer :
Let's begin by understanding the given data and subsequently determining the correlation and causation.
### Given Data:
We have the radii and circumference measurements of several objects:
```
Radius (in.) Circumference (in.)
3 18.8
4 25.1
6 37.7
9 56.5
```
### Correlation:
The correlation coefficient measures the strength and direction of a linear relationship between two variables. The correlation coefficient `r` lies between -1 and 1:
- `r = 1` indicates a perfect positive correlation.
- `r = -1` indicates a perfect negative correlation.
- `r = 0` indicates no correlation.
In this case, the correlation coefficient for the radii and circumferences is approximately `0.9999989948784027`.
### Interpretation of the Correlation Coefficient:
- A correlation coefficient greater than 0.7 generally indicates a strong positive correlation.
- Since `0.9999989948784027` is very close to 1, this tells us that there is a strong positive correlation between the radius and circumference of the objects.
### Causation:
Causation implies that changes in one variable directly cause changes in another variable. Given the relationship between radius and circumference:
- The circumference of a circle is directly related to its radius through the formula [tex]\( C = 2\pi r \)[/tex].
- This indicates a direct and causal relationship between radius and circumference.
### Conclusion:
- The strength of the correlation is strong since the correlation coefficient is very close to 1.
- The relationship between radius and circumference is likely causal as circumference is directly related to radius in a circle.
Therefore, the best description of the relationship is:
It is a strong positive correlation, and it is likely causal.
### Given Data:
We have the radii and circumference measurements of several objects:
```
Radius (in.) Circumference (in.)
3 18.8
4 25.1
6 37.7
9 56.5
```
### Correlation:
The correlation coefficient measures the strength and direction of a linear relationship between two variables. The correlation coefficient `r` lies between -1 and 1:
- `r = 1` indicates a perfect positive correlation.
- `r = -1` indicates a perfect negative correlation.
- `r = 0` indicates no correlation.
In this case, the correlation coefficient for the radii and circumferences is approximately `0.9999989948784027`.
### Interpretation of the Correlation Coefficient:
- A correlation coefficient greater than 0.7 generally indicates a strong positive correlation.
- Since `0.9999989948784027` is very close to 1, this tells us that there is a strong positive correlation between the radius and circumference of the objects.
### Causation:
Causation implies that changes in one variable directly cause changes in another variable. Given the relationship between radius and circumference:
- The circumference of a circle is directly related to its radius through the formula [tex]\( C = 2\pi r \)[/tex].
- This indicates a direct and causal relationship between radius and circumference.
### Conclusion:
- The strength of the correlation is strong since the correlation coefficient is very close to 1.
- The relationship between radius and circumference is likely causal as circumference is directly related to radius in a circle.
Therefore, the best description of the relationship is:
It is a strong positive correlation, and it is likely causal.