Questions: Given the least-squares regression line Y hat = 5 - 2X, what may be said about the relationship between the two variables, X and Y? - The relationship between x and y is positive. - The relationship between x and y is negative. - As X increases, so does Y. - The mean of the independent variable for all levels of the independent variable can be connected by a straight line.

Solution Steps

To determine the relationship between the two variables \(X\) and \(Y\) given the least-squares regression line \(\hat{Y} = 5 - 2X\), we need to analyze the slope of the regression line. The slope indicates the direction and strength of the relationship between \(X\) and \(Y\).

1. Identify the slope from the regression equation.
2. Determine if the slope is positive or negative.
3. Conclude the nature of the relationship based on the sign of the slope.

Given the least-squares regression line: \[ \hat{Y} = 5 - 2X \] The intercept is \(5\) and the slope is \(-2\).

The slope of the regression line is \(-2\). Since the slope is negative, this indicates a negative relationship between the variables \(X\) and \(Y\).

A negative slope means that as \(X\) increases, \(Y\) decreases. Therefore, the relationship between \(X\) and \(Y\) is negative.

Final Answer