 |
| The
Simple Linear Regression
Regression analysis is a statistical tool that utilizes the relation
between two or more quantitative variables so that one variable (dependent
variable) can be predicted from the others (independent variables). For
example, if one knows the relationship between advertising expenditures and
sales, one can predict sales by regression analysis once the level of
advertising expenditures has been set. In this chapter, we specifically
consider the case when a single independent variable is used for predicting
the dependent variable and the dependent variable and the independent
variable are linearly related.
|
| 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
- One should only do prediction within the range where you observed X.
- Sum of the residuals equals zero.
- 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)
| Inferences
for Regression Parameters
|
In many situations, a general form for a 
% confidence interval for a parameter 
is [ ^
(critical value)SE(^
), ] where 
is a sample statistic used to estimate
, SE 
[the standard error of
] gives the variation of
and the critical value is a value such as 
or  .
Thus, all we have to do is (1) find the formula for SE
and (2) ``plug-in'' numbers into this general equation to get a confidence
interval. The following confidence (prediction) intervals all follow this
rule.
|
| | | | | |
is the error (also called random deviation).
,
.
Therefore, 
.
.
(thousand).
