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Question
Given the following information about the production and demand of a commodity obtain the two regression lines:
| X | Y | |
| Mean | 85 | 90 |
| S.D. | 5 | 6 |
The coefficient of correlation between X and Y is 0.6. Also estimate the production when demand is 100.
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Solution
Given, `bar x = 85, bar y = 90, sigma_"X" = 5, sigma_"Y" = 6`, r =0.6
`"b"_"YX" = "r" sigma_"Y"/sigma_"X" = 0.6 xx 6/5 = 0.72`
`"b"_"XY" = "r" sigma_"X"/sigma_"Y" = 0.6 xx 5/6 = 0.5`
The regression equation of Y on X is
`("Y" - bar y) = "b"_"YX" ("X" - bar x)`
(Y - 90) = 0.72 (X - 85)
Y - 90 = 0.72 X - 61.2
Y = 0.72X - 61.2 + 90
Y = 28.8 + 0.72 X ....(i)
The regression equation of X on Y is
`("X" - bar x) = "b"_"XY" ("Y" - bar y)`
(X - 85) = 0.5(Y - 90)
X - 85 = 0.5 Y - 45
X = 0.5 Y - 45 + 85
X = 40 + 0.5Y ....(ii)
For Y = 100, from equation (ii) we get
X = 40 + 0.5(100) = 40 + 50 = 90
∴ The production is 90 when demand is 100.
Notes
The answer in the textbook is incorrect.
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byx = `square/square`
Regression equation of x on y is `(x - barx) = "b"_(xy) (y - bary)`
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Regression equation of y on x is `(y - bary) = "b"_(yx) (x - barx)`
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Mean of y = 28
Regression coefficient of y on x = – 1.2
Regression coefficient of x on y = – 0.3
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`y - square = square (50 - square)`
∴ y = `square`
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`x - square = square (25 - square)`
∴ x = `square`
Mean of x = 25
Mean of y = 20
`sigma_x` = 4
`sigma_y` = 3
r = 0.5
byx = `square`
bxy = `square`
when x = 10,
`y - square = square (10 - square)`
∴ y = `square`
bXY . bYX = ______.
