Given the following information about the production and demand of a commodity. Obtain the two regression lines: Production(X) Demand(Y) Mean 85 90 Variance 25 36 Coefficient of correlation between - Mathematics and Statistics

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Sum

Given the following information about the production and demand of a commodity.

Obtain the two regression lines:

  Production
(X)
Demand
(Y)
Mean 85 90
Variance 25 36

Coefficient of correlation between X and Y is 0.6. Also estimate the demand when the production is 100 units.

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Solution

Given, `bar(x)` = 85, `bar(y)` = 90, `sigma_x^2` = 25, `sigma_y^2` = 36, r = 0.6

∴ `sigma_x` = 5, `sigma_y` = 6

byx = `"r" sigma_y/sigma_x = 0.6 xx 6/5` = 0.72

bxy = `"r" sigma_x/sigma_y = 0.6 xx 5/6` = 0.5

The regression equation of Y on X is given by `("Y" - bary) = "b"_(xy)  ("X" - barx)`

(Y – 90) = 0.72(X – 85)

Y – 90 = 0.72X – 61.2

Y = 0.72X – 61.2 + 90

Y = 28.8 + 0.72X     ......(i)

The regression equation of X on Y is given by `("X" - barx) = "b"_(xy)  ("Y" - bary)`

(X – 85) = 0.5(Y – 90)

X – 85 = 0.5Y – 45

X = 0.5Y – 45 + 85

X = 40 + 05Y     ......(ii)

For X = 100, from equation (i) we get

Y = 28.8 + 0.72(100)

= 28.8 + 72

= 100.8

∴ The production is 90 when demand is 100.

Concept: Properties of Regression Coefficients
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Chapter 2.3: Linear Regression - Q.4

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x y `x - barx` `y - bary` `(x - barx)(y - bary)` `(x - barx)^2` `(y - bary)^2`
1 5 – 2 – 4 8 4 16
2 7 – 1 – 2 `square` 1 4
3 9 0 0 0 0 0
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bxy = `square/square`

byx = `square/square`

∴ Regression equation of x on y is `square`

∴ Regression equation of y on x is `square`


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