Question

The coefficient of correlations is r between X and Y having standard deviation σx and σy. The tangent of the between two lines of regression is:
1. \(\dfrac{1-r^2}{r}\)
2. \(\dfrac{1-r^2}{r} \sigma x \sigma y\)
3. \(\dfrac{1-r^2}{r} \dfrac{\sigma x \sigma y}{\sigma x + \sigma y}\)
4. \(\dfrac{1-r^2}{r} \dfrac{\sigma x \sigma y}{\sigma x^2 + \sigma y^2}\)

Answer 1

Correct Answer - Option 4 : \(\dfrac{1-r^2}{r} \dfrac{\sigma x \sigma y}{\sigma x^2 + \sigma y^2}\)

Given

Coefficient of correlation r between X and Y = r(X, Y) = r_xu = r

Standard deviation for X and Y = σ_x and σ_y

Calculation

The tangent of angle between two lines are given by

⇒ (1 - r)²( σx × σ_y)/r (σ_x² + σ _y²)

Equation of line of regression Y on X is

⇒ Y - y̅ = r × σ_y(X - x̅)/σ_x

The equation of line of regression X on Y is 

⇒ ⇒ X - x̅ = r × σ_x(Y - y̅ )/σ_y

Here slope of the line are

⇒ r × σ_y/σ_x and r × σ_x/σ_y

If θ is the angle between the two lines of regression

⇒ Tan θ =(r × σy/σ_x - r × σx/σ_x)/(1 + r × σy/σ_x × σx/σ_y)

⇒ (r² - 1)(σ_x σ_y)/(r × σ_x² × σ _y²)

⇒ Tan θ = (1 - r²)(σx σy)/(r × σx2 × σ y2)

⇒ θ = tan^-1{ (1 - r2)(σx σy)/(r × σx2 × σ y2)

Whenever two lines intersect, there are two angles between them, one is acute and another obtuse angle

If tan θ > 0 or o < θ < π/2 i.e θ is acute angle

.If  tan θ < 0 or π/2 < θ < 0 i.e θ is obtuse angle

∴ 0 < r² < 1, the acute angle (θ₁) and Obtuse angle (θ₂) between two lines of regression are given by

θ₁ = tan-1{ (1 - r2)(σx σy)/(r × σx2 × σ y2)

θ₂ = tan-1{ (r² - 1)(σx σy)/(r × σx2 × σ y2)