If two regression lines are x+3y=7 and 2x+5y=12 then x and y are respectively

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The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.

Find:

(a) Correlation coefficient

(b) `sigma_x/sigma_y`

We assume that 2x + 3y - 6 = 0 to be the line of regression of y on x. 

2x + 3y - 6 = 0

⇒ `x = - 3/2y + 3`

⇒ `"bxy" = - 3/2`

5x + 7y - 12 = 0 to be the line of regression of x on y.

5x + 7y - 12 = 0

⇒ `y = - 5/7x + 12/7`

⇒  `"byx" = - 5/7`

Now,

r = `sqrt("bxy.byx") = sqrt(15/14)`

byx = `(rσ_y)/(σ_x) = - 5/7, "bxy" = (rσ_x)/(σ_y) = - 3/2`

⇒ `(σ_x^2)/(σ_y^2) =  (3/2)/(5/7)`

⇒ `(σ_x^2)/(σ_y^2) = 21/10`

⇒ `(σ_x)/(σ_y) = sqrt(21/10)`.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression

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Answer (Detailed Solution Below)

Option 3 :

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10 Qs. 10 Marks 10 Mins

Concept:

The line of regression of y on x is given by:  where byx is called the regression coefficient of y on x.

Similarly, the line of regression of x on y is given by:  wherebxy is called the regression coefficient of x on y.

The correlation coefficient r2 = byx × bxy

The two lines of regression intersect each other at 

Calculation:

Given: Two regression lines are 6x + y = 30 and 3x + 2y = 25.

As we know that, the two lines of regression intersect each other at 

By solving these two equations: 6x + y = 30 and 3x + 2y = 25

We get 

We can write 6x + y = 30 as line of regression of x on y:    ------(1)

By comparing equation (1), with line of regression of x on y which is given by:  we get  

Similarly, we can write 3x + 2y = 25 as line of regression of y on x:     ------(2)

By comparing equation (2), with line of regression of x on y which is given by : we get 

As we know that, r2 = byx × bxy

As we know that, sign of 

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