In a regression and correlation analysis, if $r^2$=1, then which of the following statements is true?

• A. There is no linear relationship between the variables.
• B. The correlation coefficient $r$ must be $0$.
• C. All data points lie exactly on the regression line.
• D. The regression line has a slope of $0$.

Answer: (C)

Verified step by step guidance

1. Recall that the coefficient of determination, denoted as \(r^2\), measures the proportion of the variance in the dependent variable that is predictable from the independent variable.
2. Understand that when \(r^2 = 1\), it means 100% of the variance in the dependent variable is explained by the regression model, indicating a perfect fit.
3. Recognize that a perfect fit implies all data points lie exactly on the regression line, meaning there is a perfect linear relationship between the variables.
4. Note that if \(r^2 = 1\), the correlation coefficient \(r\) must be either +1 or -1, not zero, because zero correlation means no linear relationship.
5. Conclude that the regression line does not necessarily have a slope of zero; instead, the slope corresponds to the exact linear relationship that fits all points perfectly.