Question

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

Short answer

The sign of the slope \(b\) of the least-squares line is always the same as the sign of the sample correlation coefficient \(r\) because both measure the direction of the linear relationship between two variables. A positive sign indicates y generally increases as x increases, while a negative sign indicates y generally decreases as x increases.

Step-by-step solution

- Definition of Correlation Coefficient

The correlation coefficient (\(r\)) measures the strength and direction of a linear relationship between two variables. The value of the correlation coefficient varies between -1 and 1, where -1 indicates a perfect negative linear relationship, 1 a perfect positive linear relationship, and 0 no linear relationship. The sign of the correlation coefficient indicates the direction of the relationship.

- Explanation of the Least-Squares Line

The least-squares line, also known as the regression line, represents the best-fit line that minimizes the sum of the squared differences (or errors) between the observed and predicted values. The slope of this line, denoted here as \(b\), will be positive if, in general, y increases as x increases and negative if y decreases as x increases.

- Relation Between the Signs of b and r

Since both the correlation coefficient and the slope of the least-squares line indicate the direction of a linear relationship, their signs will always be the same. If the correlation coefficient is positive, indicating a positive linear relationship, the slope of the least-squares line will also be positive. Conversely, if the correlation coefficient is negative, indicating a negative linear relationship, the slope of the least-squares line will also be negative.