Which of the following is the formula for the y-intercept of the "best-fit" line equation? \[ \begin{array}{l} b=r\left(\frac{s_{y}}{s_{z}}\right) \\ b=\bar{y}-m \bar{x} \\ b=\bar{x}-m \bar{y} \\ b=y-m x \\ b=m \bar{x}-\bar{y}\end{array} \]

To find the formula for the y-intercept \( b \) of the "best-fit" line equation, we need to understand the components involved in the equation of a line in the context of linear regression. The equation of a best-fit line (linear regression line) can be expressed as: \[ y = mx + b \] where: - \( m \) is the slope of the line, - \( b \) is the y-intercept, - \( \bar{x} \) is the mean of the x-values, - \( \bar{y} \) is the mean of the y-values. The formula for the y-intercept \( b \) can be derived from the equation of the line. It can be expressed as: \[ b = \bar{y} - m \bar{x} \] This means that the y-intercept is equal to the mean of the y-values minus the product of the slope and the mean of the x-values. Now, let's analyze the options provided: 1. \( b = r\left(\frac{s_{y}}{s_{z}}\right) \) - This is not the formula for the y-intercept. 2. \( b = \bar{y} - m \bar{x} \) - This is the correct formula for the y-intercept. 3. \( b = \bar{x} - m \bar{y} \) - This is not the formula for the y-intercept. 4. \( b = y - m x \) - This is a rearranged form of the line equation but not specifically for the y-intercept. 5. \( b = m \bar{x} - \bar{y} \) - This is incorrect. Thus, the correct answer is: \[ b = \bar{y} - m \bar{x} \]