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

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UpStudy AI · Accepted Answer

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:

$$
b = \bar{y} - m \bar{x}
$$
The y-intercept $$ b $$ of the best-fit line is $$ b = \bar{y} - m \bar{x} $$.

Which of the following is the formula for the y-intercept of the “best-fit” line equation?

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