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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
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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