Point Estimates of Parameters of the Model

Recall the following notations:

Then

• = point estimate of = .
• = point estimate of = .
• The estimated regression line is: Y = + X.
• Point estimate of the mean y value when x = is + .
• Point prediction of a new observation when x = is + .

Also recall that the i-th residual is defined to be [ e_i=y_i-y_i, where y_i= b_0 +b_1 x_i, ] and the residual sum of squares, SSRes is

We then have

• = point estimate of = SSRes/(n-2).

Note that

1. One should only do prediction within the range where you observed X.
2. Sum of the residuals equals zero.
3. The estimated slope and the sample correlation coefficient (Week 2) have the same sign. In fact = r .

EXAMPLE:\ Eddie's Restaurant decides to invest some money on advertising. To have a better idea of what the sales amount would be if they invest 5 to 7 thousand dollars on advertising, they collect the following data on the relationship between advertising X and sales Y from a sample of five restaurants (all numbers are in thousands of dollars)

• , . Therefore, .
• .
• The estimated regression line is: Y = 35.042 + 1.394 X.
• Suppose Eddie is going to put down 6 (thousand) dollars as the advertising expense, what do you predict the restaurant will earn?

  Ans: (thousand).