It can be shown that the b1 estimator can be written as It can be shown that the b1 estimator can be written as b1 = kiYi where ki = xi - x/ (xi - x)2 Show: ki = 0 kixi = 1 ki2 = 1/ (xi - x)2 Solution (a) let t=sum(i = 1 to n )(x_i-xbar)^2 hence..sum(i=1 to n)k_i =sum(i=1 To n)(x_i-xbar)/t = (1/t)sum(i=1 To n)(x_i-xbar)=(1/t)(nxbar- nxbar)=0 (b)sum(i =1to n)(x_i-xbar)*x_i =t....hence sum(i=1 to n)K_ix_i =1 (c)sum(i =1to n)K_i^2 = t/t^2 =1/t=1/sum(i =1to n) (x_i-xbar)^2.

1. It can be shown that the b1 estimator can be written as It can be shown that the b1 estimator can
be written as b1 = kiYi where ki = xi - x/ (xi - x)2 Show: ki = 0 kixi = 1 ki2 = 1/ (xi - x)2
Solution
(a) let t=sum(i = 1 to n )(x_i-xbar)^2
hence..sum(i=1 to n)k_i =sum(i=1 To n)(x_i-xbar)/t = (1/t)sum(i=1 To n)(x_i-xbar)=(1/t)(nxbar-
nxbar)=0
(b)sum(i =1to n)(x_i-xbar)*x_i =t....hence sum(i=1 to n)K_ix_i =1
(c)sum(i =1to n)K_i^2 = t/t^2 =1/t=1/sum(i =1to n) (x_i-xbar)^2