Mean, Fitted-Value, Error, and Residual in Simple Linear Regression

observation number

5

indicate:

μ

^

y

ϵ

e

random seed

5983

0

2

4

6

8

10

0

2

4

6

8

10

x

y

true line and estimated line

This Demonstration simulates the simple linear regression

y

i

=a+bx

i

+ϵ

i

,

i=1,…,10

. The red line is the plot of

y=a+bx

, while the blue line is the plot of

^

y

=

^

a

+

^

b

x

, where

^

a

and

^

b

are the least-squares estimates. The

x

's are ordered from smallest to largest and for any particular observation number

i

that is selected, a double arrow is used to indicate

μ

i

=a+bx

i

,

^

y

i

=

^

a

+

^

b

x

i

,

ϵ

i

=y

i

-μ

i

, or

e

i

=y

i

-

^

y

i

. In the thumbnail,

μ

has been selected with

i=5

, so the double arrow shows the value of

μ

5

. The numerical values of these quantities are displayed when you mouseover one of the data points. Select some other observation numbers. Can you give approximate values for the quantities shown by the arrows? Check your answer by mousing over the selected observation.