Two regression lines are 2x 3y 4 0 and x 2y 6 0 the correlation coefficient between x and y is

1. \(\bar x = \; - 0.5,\;\;\bar y = - 4.5\)
2. \(\bar x = \;0.5,\;\;\bar y = 4.5\)
3. \(\bar x = \;4.5,\;\overline {\;y} = - 0.5\)
4. \(\bar x = \; - 0.5,\;\overline {\;y} = 4.5\)

Answer (Detailed Solution Below)

Option 3 : \(\bar x = \;4.5,\;\overline {\;y} = - 0.5\)

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Electric charges and coulomb's law (Basic)

10 Questions 10 Marks 10 Mins

CONCEPT:

Two regression lines always intersect at their mean or average values (\(\bar x,\;\bar y\). In other words if we solve two regression equations we get the average values of x and y.

CALCULATIONS:

Given equations are x – y – 5 = 0 and x + y – 4 = 0

On adding both the equations 2x = 9            

⇒ x = 4.5 (Also \(\bar x\))

By substituting x = 4.5  in equation x – y – 5 = 0 we get

⇒ y = - 0.5 (Also \(\bar y\))

So, \(\bar x = 4.5 \ and\ \bar y = - 0.5\)

Hence, option C is the correct answer.

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the question is a few lines of regression r 4 x + 2 Y - 3 = 20 and 3 X + 6 Y + 5 = 20 then we have to find the correlation Coefficient so the first step will be first regression question let us suppose that if we are talking about X on Y and I'm taking the first equation for this 4 x + 2 Y - 3 = 20 and the second equation I'm taking it as y on x 3 x + X Y + 5 = 20 and the second step we are going to solve the test and Y on X so I took the first equation is x on Y so it will be like I will keep the variable x on one side and rest of the path on another site so

question will become 4 x equals to minus 2 Y + 3 X will become minus 1 by 2 Y + 3 by 4 similarly for the second equation it is we took it as y on X so it will become 6 y equals to minus 5 minus 3 x square equal become y equals to minus 1 by 2 x minus 5 by 6 so from both of the equation we can see that be x y equals to minus 1 by 2 and b y x equals to minus 1 by 2 we know the formula of correlation Coefficient witches correlation coefficient

which is represented by our that is equals to under root x y x y x = 2 under root minus 1 by 2 x minus 1 by 2 and after solving the under root we will get one by two sobek correlation Coefficient as plus minus 1 by 2 but one more thing to note but are has the same sign has the same sign as regression coefficient regression coffee Saint therefore

are is equals to minus 1 by 2 and this is your answer thank you

The given regression equations are
2x + 3y – 6 = 0 and 2x + 2y – 12 = 0

(i) Let 2x + 3y – 6 = 0 be the regression equation of Y on X

∴ The equation becomes 3Y = – 2X + 6

i.e., Y = `(-2)/3 "X" + 6/3`

Comparing it with Y = bYX X + a, we get

`"b"_"YX" = - 2/3`

Now, the other equation, i.e., 2x + 2y – 12 = 0 is the regression equation of X on Y.

∴ The equation becomes 2X = –2Y + 12

i.e., X = `- 2/2 "Y" + 12/2`

Comparing it with X = bXY Y + a' we get

`"b"_"XY" = - 2/2 = - 1`

∴ r = `+-sqrt("b"_"XY" * "b"_"YX")`

`= +- sqrt(-1 * (- 2/3)) = +-sqrt(2/3) = +- 0.82`

since bXY and bYX are negative,

r is also negative.

∴ r = - 0.82

(ii) `"b"_"XY" = "r" sigma_"X"/sigma_"Y"`

∴ `- 1 = - 0.82 xx sigma_"X"/sigma_"Y"`

∴ `sigma_"X"/sigma_"Y" = (- 1)/- 0.82`

∴ `sigma_"X"/sigma_"Y"` = 1.22

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)`.

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