Question

complete the statement. when all the points fall on the regression line, the value of the correlation coefficient is \boxed{}.

Explanation:

Step1: Recall correlation coefficient concept

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. When all data points lie exactly on the regression line, the linear relationship is perfect.

Step2: Determine the value

For a perfect positive linear relationship (all points on an increasing line), \( r = 1 \). For a perfect negative linear relationship (all points on a decreasing line), \( r=- 1 \). In either case, when all points are on the regression line, the absolute value of \( r \) is \( 1 \) (i.e., \( r = 1 \) or \( r=-1 \)). But generally, we say the correlation coefficient is \( \pm1 \), and the most concise way is to note that it's \( 1 \) or \( - 1 \), but since the problem is about the case when all points are on the line (perfect linear correlation), the value is \( \pm1 \), and typically expressed as \( 1 \) or \( - 1 \), but the standard answer here is that it's \( 1 \) or \( - 1 \) (more precisely, \( \boldsymbol{\pm1} \)). However, in the context of the question, the key is that when all points lie on the regression line, the correlation coefficient is \( 1 \) (for positive perfect correlation) or \( - 1 \) (for negative perfect correlation), so the value is \( \pm1 \), and often we can say \( 1 \) or \( - 1 \), but the boxed answer here should be \( \pm1 \) or \( 1 \) / \( - 1 \). But following the problem's context, the answer is \( \pm1 \) (or \( 1 \) or \( - 1 \)).

Answer:

\(\pm 1\) (or \(1\) or \(-1\))