Which of the following statements concerning correlation analysis is not true?

• A. A correlation coefficient of $0$ indicates that there is no linear relationship between two variables.
• B. The sign of the correlation coefficient indicates the direction of the linear relationship.
• C. A high correlation coefficient implies that changes in one variable cause changes in the other variable.
• D. The correlation coefficient is always between $-1$ and $1$.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand what a correlation coefficient represents. It measures the strength and direction of a linear relationship between two variables.
2. Step 2: Recall that a correlation coefficient of 0 means there is no linear relationship between the variables, but it does not imply no relationship at all (there could be a non-linear relationship).
3. Step 3: Recognize that the sign of the correlation coefficient (positive or negative) indicates the direction of the linear relationship: positive means both variables increase together, negative means one increases as the other decreases.
4. Step 4: Remember that the correlation coefficient always lies between -1 and 1, where -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and values closer to 0 indicate weaker linear relationships.
5. Step 5: Identify the incorrect statement: a high correlation coefficient does not imply causation. Correlation measures association, but it does not prove that changes in one variable cause changes in the other.