Question

Line of Regression of y on x

• A. \( y - \bar{y} = \frac{\sigma_y}{\sigma_x} (x - \bar{x}) \)
• B. \( y - \bar{y} = r \frac{\sigma_y}{\sigma_x} (x - \bar{x}) \)
• C. \( y - \bar{y} = \frac{1}{r} \frac{\sigma_y}{\sigma_x} (x - \bar{x}) \)
• D. \( x - \bar{x} = r \frac{\sigma_x}{\sigma_y} (y - \bar{y}) \)

Hint:

Regression Lines. Line of y on x: \( (y - \bar{y) = b_{yx (x - \bar{x) \), where slope \( b_{yx = r (\sigma_y / \sigma_x) \). Line of x on y: \( (x - \bar{x) = b_{xy (y - \bar{y) \), where slope \( b_{xy = r (\sigma_x / \sigma_y) \). Note that \(r^2 = b_{yx \times b_{xy\).

Answer 1

The Correct Option is B

The equation for the line of regression of y on x is used to predict the value of y for a given value of x
It represents the best linear fit minimizing the sum of squared errors in the y-direction
The equation passes through the mean point \((\bar{x}, \bar{y})\) and has a slope \(b_{yx}\)
The equation is given by: $$ (y - \bar{y}) = b_{yx} (x - \bar{x}) $$ The regression coefficient \(b_{yx}\) (slope of y on x) is calculated as: $$ b_{yx} = r \frac{\sigma_y}{\sigma_x} $$ where \(r\) is the Pearson correlation coefficient between x and y, \(\sigma_y\) is the standard deviation of y, and \(\sigma_x\) is the standard deviation of x
Substituting the expression for the slope gives: $$ y - \bar{y} = r \frac{\sigma_y}{\sigma_x} (x - \bar{x}) $$ This matches option (2)
Option (4) represents the regression line of x on y
Options (1) and (3) have incorrect slopes