Explanations:The least-squares regression line is a statistical tool used to model the relationship between two variables, typically denoted as \(x\) (the independent variable) and \(y\) (the dependent variable). In this context, the independent variable is the arm span (\(x\)), and the dependent variable is the height (\(y\)). The equation of the least-squares regression line is generally written as:
\[ y = mx + b \]
where \(m\) is the slope of the line and \(b\) is the y-intercept. The slope \(m\) is a crucial component of the regression line because it indicates the direction and strength of the relationship between the two variables. Specifically, the slope tells us how much the dependent variable \(y\) changes for a one-unit increase in the independent variable \(x\).
In the context of the problem, the least-squares regression line is used to predict height (\(y\)) from arm span (\(x\)). The slope of this line is positive, which means that as arm span increases, height also tends to increase. This positive relationship between arm span and height indicates a positive correlation.
Correlation is a measure of the strength and direction of the linear relationship between two variables. The correlation coefficient, denoted as \(r\), ranges from -1 to 1. A positive correlation coefficient (greater than 0) indicates a positive relationship, where both variables tend to increase together. A negative correlation coefficient (less than 0) indicates a negative relationship, where one variable tends to decrease as the other increases. A correlation coefficient of 0 indicates no linear relationship.
Given that the slope of the least-squares regression line is positive, we can conclude that the correlation between height and arm span is also positive. Therefore, the correct answer is that the correlation is greater than zero.
It is important to note that while the slope of the regression line indicates the direction of the relationship, it does not directly provide the exact value of the correlation coefficient. The correlation coefficient is a standardized measure that takes into account the variability of both variables, and it ranges from -1 to 1. The slope, on the other hand, is specific to the units of the variables and does not have a standardized range.
In summary, the positive slope of the least-squares regression line indicates a positive correlation between height and arm span, meaning the correlation is greater than zero.