Question

the radius and circumference of several objects were measured.
radius and circumference of objects
radius (in.) | circumference (in.)
3 | 18.8
4 | 25.1
6 | 37.7
9 | 56.5
which best describes the strength of the correlation, and what is true about the causation between the variables?
it is a weak positive correlation, and it is not likely causal.
it is a weak positive correlation, and it is likely causal.
it is a strong positive correlation, and it is likely causal.

Explanation:

Step1: Recall the formula for circumference

The formula for the circumference \( C \) of a circle is \( C = 2\pi r \), where \( r \) is the radius. This is a linear relationship with a positive slope, meaning as radius increases, circumference should increase in a very predictable way.

Step2: Analyze the correlation strength

Looking at the data: when radius is 3, circumference is 18.8; radius 4, circumference 25.1; radius 6, circumference 37.7; radius 9, circumference 56.5. Let's check the ratio \( \frac{C}{r} \) for each: \( \frac{18.8}{3}\approx6.27 \), \( \frac{25.1}{4}\approx6.28 \), \( \frac{37.7}{6}\approx6.28 \), \( \frac{56.5}{9}\approx6.28 \). These are very close to \( 2\pi\approx6.28 \), so the points lie very close to a straight line, indicating a strong positive correlation.

Step3: Analyze causation

Since the circumference of a circle is directly determined by its radius (via \( C = 2\pi r \)), changing the radius causes a change in the circumference. So there is a causal relationship.

Answer:

It is a strong positive correlation, and it is likely causal. (Note: Assuming the third option (not fully shown but from analysis) is "It is a strong positive correlation, and it is likely causal" as the data follows \( C = 2\pi r \) closely, showing strong positive correlation and causal link because radius determines circumference.)