Which of the following statements about the correlation coefficient is true?

• A. A correlation coefficient of $-0.8$ indicates a strong positive linear relationship.
• B. A correlation coefficient of $0$ indicates no linear relationship between two variables.
• C. The correlation coefficient can only take values between $0$ and $1$.
• D. A correlation coefficient of $1$ means there is a perfect negative linear relationship.

Answer: (B)

Verified step by step guidance

1. Recall that the correlation coefficient, often denoted as \(r\), measures the strength and direction of a linear relationship between two variables.
2. Understand the range of the correlation coefficient: it can take any value between \(-1\) and \$1\(, inclusive. This means \)-1 \leq r \leq 1$.
3. Interpret the sign and magnitude of \(r\): a positive value indicates a positive linear relationship, a negative value indicates a negative linear relationship, and a value close to zero indicates little to no linear relationship.
4. Analyze each statement: - A correlation coefficient of \(-0.8\) indicates a strong negative linear relationship, not positive. - A correlation coefficient of \$0\( indicates no linear relationship. - The correlation coefficient cannot be only between \)0\( and \)1\(; it includes negative values as well. - A correlation coefficient of \)1$ means a perfect positive linear relationship, not negative.
5. Conclude that the true statement is: 'A correlation coefficient of \$0$ indicates no linear relationship between two variables.'