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Ryan gathered data about the age of the different dogs in his neighborhood and the length of their tails.

Lengths of Tails for Dogs of Different Ages

\begin{tabular}{|c|c|}
\hline
Age (years) & Length of Tail (in.) \\
\hline
2 & 12 \\
\hline
3 & 0 \\
\hline
6 & 7 \\
\hline
10 & 4 \\
\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 negative correlation, and it is not likely causal.
B. It is a weak negative correlation, and it is likely causal.
C. It is a strong negative correlation, and it is not likely causal.
D. It is a strong negative correlation, and it is likely causal.

Sagot :

GinnyAnswer
To determine the strength and direction of the correlation between the age of the dogs and the length of their tails, we will analyze the Pearson correlation coefficient. Here is the step-by-step approach:

1. Data Splitting:
- The given data can be split into two arrays:
- Age (years): [tex]\([2, 3, 6, 10]\)[/tex]
- Length of Tail (inches): [tex]\([12, 0, 7, 4]\)[/tex]

2. Calculating Pearson Correlation Coefficient:
- The Pearson correlation coefficient [tex]\( r \)[/tex] quantifies the strength and direction of the relationship between two variables.
- The value of [tex]\( r \)[/tex] ranges from [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex]:
- [tex]\( r > 0 \)[/tex] suggests a positive correlation (as one variable increases, the other also increases).
- [tex]\( r < 0 \)[/tex] suggests a negative correlation (as one variable increases, the other decreases).
- The magnitude of [tex]\( r \)[/tex] (i.e., [tex]\( |r| \)[/tex]) near [tex]\( 1 \)[/tex] indicates a strong correlation, while near [tex]\( 0 \)[/tex] indicates a weak correlation.

3. Interpreting the Result:
- The calculated Pearson correlation coefficient [tex]\( r \)[/tex] for the given data is approximately [tex]\( -0.2705 \)[/tex].

4. Assessing the Strength of the Correlation:
- The magnitude of [tex]\( r \)[/tex] (i.e., [tex]\( |r| = 0.2705 \)[/tex]) suggests that the correlation is weak (since it is closer to 0 than to 1).

5. Determining the Direction of the Correlation:
- Since [tex]\( r = -0.2705 \)[/tex] (a negative value), the correlation is negative.

6. Drawing a Conclusion about Causation:
- Correlation alone does not imply causation. Just because two variables are correlated does not mean one variable causes changes in the other. Other factors may influence both variables, or it could be a coincidence.
- In this context, the weak negative correlation between the age of the dogs and the length of their tails does not provide strong evidence of a causal relationship.

Putting all this together:

- The strength of the correlation is weak.
- The direction of the correlation is negative.
- It is not likely that there is a causal relationship between the age of the dogs and the length of their tails, given the weak correlation.

Therefore, the best description is:

It is a weak negative correlation, and it is not likely causal.
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