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Question
From the data 7 pairs of observations on X and Y following results are obtained :
∑(xi - 70) = - 38 ; ∑ (yi - 60) = - 5 ;
∑ (xi - 70)2 = 2990 ; ∑ (yi - 60)2 = 475 ;
∑ (xi - 70) (yi - 60) = 1063
Obtain :
(a) The line of regression of Y on X.
(b) The line of regression of X on Y.
Solution
Let ui = xi - 70 = -38
vi = yi - 60 = -5
Assumed mean of variable X and Y are 70 and 60 respectively,
Now ∑ ui = - 38 ∑ vi = - 5
`sum "u"_"i"^2 = 2990` `sum "v"_"i"^2 = 475`
∑ ui vi = 1063 n = 7
Cov (x , y) = Cov (u , v)
`= (sum "u"_"i""v"_"i")/"n" - bar "u"_"i" bar "v"_"i"`
`= (sum "u"_"i" "v"_"i")/"n" - ((sum "u"_"i")/"n" . (sum "v"_"i")/"n")`
`= 1063/7 - [(-38/7) . (-5/7)]`
= 151.857 - 3.878
= 147.98
`sigma_"x"^2 = sigma_"u"^2 = (sum u_"i"^2)/"n" - (bar u)^2`
`= (sum u_"i"^2)/7 - ((sum u_"i"^2)/"n")^2`
`= 2990/7 - (-38/7)^2`
= 427.143 - 29.469
= 397.674
`sigma_"y"^2 = sigma_"v"^2 = (sum v_"i"^2)/"n" - ((sum v_"i"^2)/"n")^2`
`= 475/7 - (-5/7)^2`
`= 3300/49`
= 67.347
`"b"_"yx" = "b"_"vu" = ("Cov"("u , v"))/ sigma_"u"^2`
`= 147.98/397.647`
= 0.3721
`"b"_"xy" = "b"_"uv" = ("Cov"("u , v"))/ sigma_"v"^2`
`= 147.98/67.347`
= 2.1973
`bar u = -38/7 = - 5.428`
`bar v = -5/7 = - 0. 714`
`therefore bar x = bar u + 70 = -5.428 + 70 = 64.572`
`bar y = bar v + 60 = - 0.714 + 60 = 59.286`
Equation of line y on x
`y - bar y = "b"+"yx" (x - bar x)`
y - 59.286 = 0.3721 (x - 64.572)
y = 0.3721 x + 35.259
Equation of line x on y
`x - bar x = "b"_"xy" (y - bar y)`
x - 64.572 = 2.1973 (y - 59.286)
x = 2.1973 y - 65.697
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