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The radius and circumference of several objects were measured.

Radius and Circumference of Objects

\begin{tabular}{|c|c|}
\hline Radius (in.) & Circumference (in.) \\
\hline 3 & 18.8 \\
\hline 4 & 25.1 \\
\hline 6 & 37.7 \\
\hline 9 & 56.5 \\
\hline
\end{tabular}

Which best describes the strength of the correlation, and what is true about the causation between the variables?

A. It is a weak positive correlation, and it is not likely causal.
B. It is a weak positive correlation, and it is likely causal.
C. It is a strong positive correlation, and it is not likely causal.
D. It is a strong positive correlation, and it is likely causal.



Answer :

GinnyAnswer
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.

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