Question

Angle between two line of regression is given by.

• A. ${\tan }^{-1}[b_{xy}–(1/b_{yx}\left)\right]/[1–(b_{xy}/b_{yx}\left)\right]$
• B. ${\tan }^{-1}[b_{yx}.b_{xy}–1]/\left[\right(b_{yx}+b_{xy}\left)\right]$
• C. ${\tan }^{-1}[b_{xy}–(1/b_{yx}\left)\right]/[1+(b_{xy}/b_{yx}\left)\right]$
• D. ${\tan }^{-1}[b_{yx}–b_{xy}]/[1+(b_{yx}.b_{xy}\left)\right]$

Answer 1

The correct option is B

${\tan }^{-1}[b_{yx}.b_{xy}–1]/\left[\right(b_{yx}+b_{xy}\left)\right]$

Explanation for the correct option:

Consider general equation line is given by,

$y=b_{yx}x+c_1\&x=b_{xy}y+c_2$

Now,

$\begin{array}{rcl}y& =& b_{\mathrm{yx}}.x+c_1\\ & \Rightarrow & m1=b_{\mathrm{yx}}\\ x& =& b_{\mathrm{xy}}.y+c_2\\ & \Rightarrow & m2=\frac{1}{{\mathrm{b}}_{\mathrm{xy}}}\\ \tan \left(\mathrm{\theta }\right)& =& \left(\frac{\mathrm{m}1-\mathrm{m}2}{1+\mathrm{m}1.\mathrm{m}2}\right)where,m_1\&m_{2}aretheslope\\ & =& \left(\frac{{\mathrm{b}}_{\mathrm{yx}}-{\displaystyle \frac{1}{{\mathrm{b}}_{\mathrm{xy}}}}}{1+{\mathrm{b}}_{\mathrm{yx}}.{\displaystyle \frac{1}{{\mathrm{b}}_{\mathrm{xy}}}}}\right)\\ & =& \left(\frac{{\displaystyle \frac{{\mathrm{b}}_{\mathrm{xy}}.{\mathrm{b}}_{\mathrm{yx}}-1}{{\mathrm{b}}_{\mathrm{xy}}}}}{{\displaystyle \frac{{\mathrm{b}}_{\mathrm{xy}+}{\mathrm{b}}_{\mathrm{yx}}}{{\mathrm{b}}_{\mathrm{xy}}}}}\right)\\ & =& \left(\frac{{\mathrm{b}}_{\mathrm{xy}}.{\mathrm{b}}_{\mathrm{yx}}-1}{{\mathrm{b}}_{\mathrm{xy}+}{\mathrm{b}}_{\mathrm{yx}}}\right)\\ \tan \left(\mathrm{\theta }\right)& =& \left(\frac{{\mathrm{b}}_{\mathrm{xy}}.{\mathrm{b}}_{\mathrm{yx}-}1-}{{\mathrm{b}}_{\mathrm{xy}+}{\mathrm{b}}_{\mathrm{yx}}}\right)\\ \theta & =& {\tan }^{-1}\left(\left(\frac{{\mathrm{b}}_{\mathrm{xy}}.{\mathrm{b}}_{\mathrm{yx}}-1}{{\mathrm{b}}_{\mathrm{xy}+}{\mathrm{b}}_{\mathrm{yx}}}\right)\right)\end{array}$

The regression of the two lines between the angle is ${\tan }^{-1}\frac{[{\mathrm{b}}_{\mathrm{yx}}.{\mathrm{b}}_{\mathrm{xy}}-1]}{\left[\left({\mathrm{b}}_{\mathrm{yx}}+{\mathrm{b}}_{\mathrm{xy}}\right)\right]}$

Hence option(B) is correct.