Suppose rxy is the correlation coefficient between two variables X and Y
where s.d.(X) = s.d.(Y). If θ is the angle between the two regression lines of Y on X and X on Y then:

This question was previously asked in
SSC CGL Tier-II ( JSO ) 2016 Official Paper ( Held On : 2 Dec 2016 )
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  1. \(sin \theta=\sqrt{\dfrac{1-r^2_{xy}}{1+r^2_{xy}}}\)
  2. \(cos \theta=\sqrt{\dfrac{1-r^2_{xy}}{1+r^2_{xy}}}\)
  3. \(tan \theta=\sqrt{1-r^2_{xy}}\)
  4. \(tan \theta=\pm\dfrac{1-r^2_{xy}}{r^2_{xy}}\)

Answer (Detailed Solution Below)

Option 1 : \(sin \theta=\sqrt{\dfrac{1-r^2_{xy}}{1+r^2_{xy}}}\)
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Detailed Solution

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Formula

If θ be the angle between the regression lines then,

Tan θ = [(1 - r2)/r] × (σx ×  σy)/(σx2 + σy2)

Explanation

According to question

⇒ σx = σy

⇒ Tanθ = [(1 - r2/r] × (σx2/(2σx)

⇒ Tanθ = [(1 - r2/r] × 1/2

⇒ Tan θ = (1 - r2/2r

⇒ Tanθ = Sinθ/cosθ

⇒ Tanθ = Sinθ × cosθ 

⇒ Sinθ = Tanθ /secθ 

⇒ Sinθ = Tanθ/√(1 + tan2θ)

⇒ Sinθ = [(1 - r2/r]/√(1 + (1 - r2/2r)2)

⇒ Sinθ = (1 - r2)/√[(1 - r2)2 + 4r2]

∴ \(sin \theta=\sqrt{\dfrac{1-r^2_{xy}}{1+r^2_{xy}}}\)

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