Question

From the data 7 pairs of observations on X and Y following results are obtained :

∑(x_i - 70) = - 38 ; ∑ (y_i - 60) = - 5 ;

∑ (x_i - 70)2 = 2990 ; ∑ (y_i - 60)2 = 475 ;

∑ (x_i - 70) (y_i - 60) = 1063

Obtain :

(a) The line of regression of Y on X.

(b) The line of regression of X on Y. 

Answer 1

Let u_i = x_i - 70 = -38 

v_i = y_i - 60 = -5 

Assumed mean of variable X and Y are 70 and 60 respectively, 

Now ∑ u_i = - 38                            ∑ v_i = - 5

`sum "u"_"i"^2 = 2990`                          `sum "v"_"i"^2 = 475`

∑ u_i v_i = 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