The correct answer is:

C. No, because while there is no linear correlation, there may be a relationship that is not linear.

Explanation:

1. The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables.
2. If $r = 0$, it indicates no linear correlation, meaning the data does not follow a linear pattern.
3. However, $r = 0$ does not rule out the possibility of a non-linear relationship. For example, there might be a curved or complex pattern that a linear correlation cannot detect.

Why other options are incorrect:

• A is incorrect because $r = 0$ indicates no linear relationship, not a perfect linear relationship.
• B is incorrect because $r$ measures both the strength and direction of a linear relationship.
• D is incorrect because $r = 0$ does not imply that the variables are completely unrelated; it only suggests they are not linearly related.

Would you like a deeper dive into this topic or have any follow-up questions?

5 Follow-Up Questions:

1. What is the range of values for the correlation coefficient $r$?
2. How can you detect non-linear relationships between two variables?
3. What methods besides correlation can be used to identify relationships between variables?
4. What does a correlation coefficient of $r = 1$ or $r = -1$ indicate?
5. Can the correlation coefficient be used for categorical data?

Tip:

When analyzing data, always look at a scatterplot first; it helps in visualizing both linear and non-linear relationships.