Question

Equations of two lines of regression are \( 2x - 5y + 33 = 0 \) and \( 3x - 9y - 108 = 0 \). Which one is the regression line of \( y \) on \( x \)?

Hint:

To distinguish between regression lines, remember that the regression line of \( y \) on \( x \) is in the form \( y = a + bx \), and the regression line of \( x \) on \( y \) is in the form \( x = a + by \).

Answer 1

We are given two equations of regression lines. The goal is to determine which one represents the regression line of \( y \) on \( x \). Step 1: Understand the form of regression equations.
The regression equation of \( y \) on \( x \) is typically in the form: \[ y = a + bx \] Where \( b \) is the regression coefficient of \( y \) on \( x \). The equation of the regression line of \( x \) on \( y \) is in the form: \[ x = a + by \] Where \( b \) is the regression coefficient of \( x \) on \( y \).
Step 2: Solve the given equations.
The first equation is: \[ 2x - 5y + 33 = 0 \] Rearrange it to get \( y \) in terms of \( x \): \[ 5y = 2x + 33 \] \[ y = \frac{2}{5}x + \frac{33}{5} \] This is of the form \( y = a + bx \), which indicates that this equation represents the regression line of \( y \) on \( x \). The second equation is: \[ 3x - 9y - 108 = 0 \] Rearrange it to get \( x \) in terms of \( y \): \[ 3x = 9y + 108 \] \[ x = 3y + 36 \] This is of the form \( x = a + by \), which indicates that this equation represents the regression line of \( x \) on \( y \).
Step 3: Conclusion.
From the above analysis, we conclude that the first equation \( 2x - 5y + 33 = 0 \) is the regression line of \( y \) on \( x \).