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If 4x – 5y + 33 = 0 and 20x – 9y = 107 are two lines of regression, then what are the values of x̅ and y̅ respectively?

If 4x – 5y + 33 = 0 and 20x – 9y = 107 are two lines of regression, then what are the values of x̅ and y̅ respectively? Correct Answer 13 and 17

Concept:

Regression lines: Let two variables x and y, if y depends on x, then the result comes in form of simple regression. 

Regression line for simple regression:

y = a + bx

Where a and b are regression parameters.

Properties:

  •  The two regression lines (of y on x and x on y) are known to intersect at a specific point (x̅, y̅).


Calculation:

Given: Regression lines,

4x – 5y + 33 = 0      ----(1)

20x – 9y = 107      ----(2)

Now, we know that, intersection point of regression lines gives a point (x̅, y̅).

Multiplying equation (1) with -5 we get,

-20x + 25y – 165 = 0      ----(3)

On adding (2) & (3) we get,

20x – 9y – 107 – 20x + 25y – 165 = 0

⇒ 16y – 272 = 0

⇒ 16y = 272

⇒ y = 17

Put y = 17 in equation (1) we get,

4x – 5(17) + 33 = 0

⇒ 4x = 85 – 33

⇒ 4x = 52

⇒ x = 13

So, x̅ = 13 and y̅ = 17.

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