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Question
For a certain bivariate data
| X | Y | |
| Mean | 25 | 20 |
| S.D. | 4 | 3 |
And r = 0.5. Estimate y when x = 10 and estimate x when y = 16
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Solution
Given, `bar x = 25, bar y = 20, sigma_"X" = 4, sigma_"Y" = 3`, r =0.5
`"b"_"YX" = "r" sigma_y/sigma_x = (0.5) 3/4 = 0.375`
`"b"_"XY" = "r" sigma_y/sigma_x = (0.5) 4/3 = 0.667`
The regression equation of Y on X is
`("Y" - bar y) = "b"_"YX" ("X" - bar x)`
(Y - 20) = 0.375 (X - 25)
Y - 20 = - 9.375 + 0.375 X
Y = 10.625 + 0.375 X
For X = 10
Y = 10.625 +0.375 × 10 = 10.625 + 3.75 = 14.375
The regression equation of X on Y is
`("X" - bar x) = "b"_"XY" ("Y" - bar y)`
(X - 25) = 0.667(Y - 20)
X - 25 = - 13.34 + 0.667 Y
X = 11.66 + 0.667 Y
For Y = 16,
X = 11.66 + 0.667(16) = 11.66 + 10.672 = 22.332
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Mean of x = 53
Mean of y = 28
Regression coefficient of y on x = – 1.2
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a. r = `square`
b. When x = 50,
`y - square = square (50 - square)`
∴ y = `square`
c. When y = 25,
`x - square = square (25 - square)`
∴ x = `square`
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Mean of x = 18
`2 square - 5 bary + 60` = 0
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| x | y | xy | x2 | y2 |
| 6 | 9 | 54 | 36 | 81 |
| 2 | 11 | 22 | 4 | 121 |
| 10 | 5 | 50 | 100 | 25 |
| 4 | 8 | 32 | 16 | 64 |
| 8 | 7 | `square` | 64 | 49 |
| Total = 30 | Total = 40 | Total = `square` | Total = 220 | Total = `square` |
bxy = `square/square`
byx = `square/square`
∴ Regression equation of x on y is `square`
∴ Regression equation of y on x is `square`
