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प्रश्न
From the data of 7 pairs of observations on X and Y, following results are obtained.
∑(xi - 70) = - 35, ∑(yi - 60) = - 7,
∑(xi - 70)2 = 2989, ∑(yi - 60)2 = 476,
∑(xi - 70)(yi - 60) = 1064
[Given: `sqrt0.7884` = 0.8879]
Obtain
- The line of regression of Y on X.
- The line regression of X on Y.
- The correlation coefficient between X and Y.
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उत्तर
Given: n =7, ∑(xi - 70) = - 35, ∑(yi - 60) = - 7,
∑(xi - 70)2 = 2989, ∑(yi - 60)2 = 476,
∑(xi - 70)(yi - 60) = 1064
Let ui = xi - 70 and vi = yi - 60
∴ ∑ ui = - 35, ∑ vi = - 7
`sum "u"_"i"^2 = 2989, sum "v"_"i"^2 = 479`
∑ ui vi = 1064
∴ `bar "u" = (sum "u"_"i")/"n" = (-35)/7 = - 5`
∴ `bar "v" = (sum "v"_"i")/"n" = (-7)/7 = - 1`
Now, `sigma_"u"^2 = (sum "u"_"i"^2)/"n" - (bar"u")^2`
`= 2989/7 - (- 5)^2` = 427 - 25 = 402
and `sigma_"v"^2 = (sum "v"_"i"^2)/"n" - (bar"v")^2`
`= 476/7 - (- 1)^2 = 68 - 1 = 67`
cov(u, v) = `(sum "u"_"i" "v"_"i")/"n" - bar"uv"`
`= 1064/7 - (- 5)(- 1)` = 152 - 5 = 147
Since the regression coefficients are independent of change of origin,
bYX = bVU and bXY = bUV
∴ bYX = bVU = `("cov" ("u", "v"))/sigma_"U"^2 = 147/402 = 0.36`
and bXY = bUV = `("cov" ("u", "v"))/sigma_"V"^2 = 147/67 = 2.19`
Also, `bar x = bar u` + 70 = - 5 + 70 = 65
and `bar y = bar v` + 60 = - 1 + 60 = 59
(i) The line of regression of Y on X is
`("Y" - bar y) = "b"_"YX" ("X" - bar x)`
∴ (Y - 59) = (0.36)(X - 65)
∴ Y - 59 = 0.36X - 23.4
∴ Y = 0.36X + 59 - 23.4
∴ Y = 0.36X + 35.6
(ii) The line of regression of X on Y is
`("X" - bar x) = "b"_"XY" ("Y" - bar y)`
∴ (X - 65) = (2.19)(Y - 59)
∴ X - 65 = 2.19Y - 129.21
∴ X = 2.19Y + 65 - 129.21
∴ X = 2.19Y - 64.21
(iii) r = `+-sqrt("b"_"YX" * "b"_"XY")`
`= +- sqrt((0.36)(2.19))`
`= +- sqrt0.7884 = +- 0.8879`
Since bYX and bXY both are positive,
r is also positive.
∴ r = 0.8879
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