Let's analyze the correlation between the age of the dogs and the length of their tails using the provided data:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Age (years)} & \text{Length of Tail (in.)} \\
\hline 2 & 12 \\
\hline 3 & 0 \\
\hline 6 & 7 \\
\hline 10 & 4 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution
1. List the age and length pairs:
- Age (years): [tex]\(2, 3, 6, 10\)[/tex]
- Length of Tail (inches): [tex]\(12, 0, 7, 4\)[/tex]
2. Calculate the correlation coefficient:
- The correlation coefficient measures the strength and direction of the linear relationship between two variables.
- The calculated correlation coefficient is approximately [tex]\(-0.2705\)[/tex].
3. Determine the type of correlation:
- A correlation coefficient of [tex]\(-0.2705\)[/tex] indicates a negative correlation because the coefficient is less than zero.
4. Determine the strength of the correlation:
- The absolute value of the correlation coefficient is [tex]\(0.2705\)[/tex].
- A correlation coefficient whose absolute value is between 0 and 0.3 (approximately) is considered a weak correlation.
- Therefore, the strength of the correlation is weak.
5. Assess causation:
- Correlation does not imply causation. Even a perfect correlation does not mean that one variable causes the other variable to change.
- Based on the weak correlation observed here, it is not likely that the variation in age causes the variation in tail length directly.
Given all of this information, the best description of the relationship between the age of the dogs and the length of their tails is:
It is a weak negative correlation, and it is not likely causal.
