If the correlation coefficient between $x$ and $y$ is equal to $-1.0$, which of the following statements is true?

• A. There is a perfect positive linear relationship between $x$ and $y$.
• B. $x$ and $y$ are not linearly related.
• C. The variables $x$ and $y$ are independent.
• D. There is a perfect negative linear relationship between $x$ and $y$.

Answer: (D)

Verified step by step guidance

1. Recall that the correlation coefficient, denoted as \(r\), measures the strength and direction of a linear relationship between two variables \(x\) and \(y\).
2. Understand that the value of \(r\) ranges from \(-1\) to \$1\(, where \)r = 1\( indicates a perfect positive linear relationship, \)r = -1\( indicates a perfect negative linear relationship, and \)r = 0$ indicates no linear relationship.
3. Given that the correlation coefficient is \(r = -1.0\), interpret this as the variables \(x\) and \(y\) having a perfect negative linear relationship, meaning as \(x\) increases, \(y\) decreases in a perfectly linear manner.
4. Recognize that this excludes the options stating no linear relationship or independence, since a perfect negative correlation implies a deterministic linear relationship between the variables.
5. Therefore, the correct interpretation is that there is a perfect negative linear relationship between \(x\) and \(y\).