[Math] help with using the “simple regression (least squares) method” of forecasting

Question

This problem is from an engineering management textbook (Morse & Babcock, 5th ed) :

2005     $48k
2006     $64k
2007     $67k
2008     $83k

"What is the sales forecast for 2009, using the simple regression (least squares) method?"

The book works through an example (shown below and not to be confused with the above), but does not give enough information for me to understand what's going on. Hopefully it won't confuse my question, but I'll give the problem they worked through here also, so that I can get help understanding how they got their numbers. They have:

Table 3-2 data was:

2005  $1100
2006  $1300
2007  $1200
2008  $1600

in b of the given problem, I can't figure where they got the parenthesized numbers–any of them! It couldn't have been by multiplying the sum of X's by the sum of Y's, or by the sum of y's. So I can't possibly do the homework problem until I understand how they are doing the give example.

Any help is appreciated.

Update: here is my work on the homework question. Can anybody give me a bit of confirmation that I've done right?

Answer 1

Does this help?

$$ \begin{align} a&=\frac{n\sum(D_iI_i)-\sum I_i\sum D_i}{n\sum I_i^2-(\sum I_i)^2}\\ &=\frac{4(0\cdot1100+1\cdot1300+2\cdot1200+3\cdot1600)-(0+1+2+3)(1100+1300+1200+1600)}{4(0^2+1^2+2^2+3^2)-(0+1+2+3)^2}\\ &=\frac{4(8500)-(6)(5200)}{4(14)-(6)^2} \end{align} $$