S= Σ (yr - kxr)^2 from r=1 to r=5 where r in the yr and xr are subscripts. By expressing S as a quadratic in k

Question

S= Σ (yr - kxr)^2 from r=1 to r=5 where r in the yr and xr are subscripts. By expressing S as a quadratic in k, show that, as k varies, S is a minimum when l is L.

The equation y=kx represents a line l through the origin of the scatter diagram. L is the regression line of y on x.

x 0 4 2 1 3
y 3 12 -1 2 14
the above is the table for the values obtained.

x 0, 4, 2, 1, 3
y 3, 12, -1, 2, 14

Answer 1

S=Σ(yr-k*xr)^2; dS/dk = 2Σ(yr-k*xr)(-xr) =0, find k; 0= -Σyr*xr + k*Σ(xr)^2 or
Σyr*xr = k*Σ(xr)^2, thence k={Σyr*xr} / {Σ(xr)^2};

==x=|==y==|==xx==|==yx==|
=00=|==3==|==00==|==00==|
=04=|=12==|==16==|==48==|
=02=|=-1==|==04==|==-2==|
=01=|=02==|==01==|==02==|
=03=|=14==|==09==|==42==|
thus k=90/30=3; y=3x; S=(3-3*0)^2 +(12-3*4)^2 +(-1-3*2)^2 +(2-3*1)^2 +(14-3*3)^2 = 84;

Answer 2

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