a) Consider the following set of regression equations:

Y₁  =   βx₁ + e₁

Y₂  =   βx₂+ e₂ ..

Y_n  =   βx_n + e_n

Suppose also that w₁, w₂, ..., w_n are a set of positive weights (known constants). Consider the function

f(β) = ∑ w_ie_i² =  ∑ w_i (y_i - βx_i)²

Find the value of β that minimizes f(β). (This value of β is called the weighted least squares estimate of β.)

b) and c) use the following random sample of n = 6 pairs of values of x and Y.

x          Y

1          3

6          7

4          12

2          5

1          4

3          5

 b) For the regression model in part a), find the (unweighted) least squares estimate of β using this data. Explain the meaning of this value.

c) Suppose that the weights described in part a) are 1,1,1,2,2,3, respectively. For the regression model in part a), find the weighted least squares estimate of β. Does it differ from your answer in b)? Can you explain why?