Sec 10.3
Coefficient of Determination and Standard
Error of the Estimate.
Review concepts
x
1
2
3
4
5
y
10
8
12
16
20
𝒚′
fill in the third row of the table for each x
value.
Review concepts
x
1
2
3
4
5
y
10
8
12
16
20
𝒚′
7.6
10.4
13.2
16
18.8
Variation
Regression
line
Bluman, Chapter 10
5
Variations
 The total variation is
calculated by:

(
y

y
)

2
This the sum of the squares of the
vertical distances from the mean.
2 parts of variation
 The total variation is made up off
two types of variation:
1. Explained variation: attributed to
the relationship between x & y.
2. Unexplained variation: due to
chance.
Explained variation

 ( y  y)
'
2
Most of the variation can be
explained by the relationship.
Unexplained variation
( y  y )
' 2
When this variation is small then the
value of r will be close to 1 or -1.
The last few slides summarized:


 ( y  y)   ( y  y)   ( y y )
2
Total variation
'
2
Explained variation
' 2
Unexplained variation
Residuals
 The values of (y-y') are called
residuals.
 A residual is the difference between
the actual value of y and the
predicted value of y' for a given x
value.
 The mean of the residuals is
always zero.
Coefficient of . . .
1.Determination,
2
r
2.non-determination,
2
1-r
Coefficient of determination, r2
The coefficient of
determination,r2, is a measure
of the variation of the
dependent variable that is
explained by the regression line
and the independent variable.
Coefficient of Determiation
The coefficient of determination is the
ratio of the explained variation to the total
variation.
 The symbol for the coefficient of
determination is r 2.


explained variation
r 
total variation

Another way to arrive at the value for r 2
is to square the correlation coefficient.
2
Bluman, Chapter 10
14
Coefficient of Nondetermination:
Coefficient nondetermination is
the measure of the rest of the
variation that is not explained
by r2.
It is the complement of r2 and
equals to 1-r2.
Coefficient of Nondetermiation
The coefficient of nondetermination is
a measure of the unexplained variation.
 The formula for the coefficient of
determination is 1.00 – r 2.

Bluman, Chapter 10
16
Some facts:
 The coefficient of determination is a
percent.
if r2=.81 that means 81% of variation
in the dependent variable is explained by
the variation in the independent variable.
 i.e.
 The coefficient of nondetermination is:
 1-81%=19% and it means that 19% …
Example:
Let r=0.9123
Find the coefficients of
determination and
nondetermination.
Explain the meaning of each.
Standard Error of the estimate


Symbol: Sest
Sest is the standard deviation of the observed y values
about the predicted y' values.
s 
est
( y  y )
n2
'
2
 y  a  y  b  xy
Sest 
n2
2
Standard Error of the Estimate

The standard error of estimate,
denoted by sest is the standard deviation
of the observed y values about the
predicted y' values. The formula for the
standard error of estimate is:
sest 
  y  y 
2
n2
Bluman, Chapter 10
20
Chapter 10
Correlation and Regression
Section 10-3
Example 10-12
Page #569
Bluman, Chapter 10
21
Example 10-12: Copy Machine Costs
A researcher collects the following data and determines
that there is a significant relationship between the age of a
copy machine and its monthly maintenance cost. The
regression equation is y  = 55.57 + 8.13x. Find the
standard error of the estimate.
Bluman, Chapter 10
22
Example 10-12: Copy Machine Costs
Machine
Age x
(years)
Monthly
cost, y
A
B
C
D
E
F
1
2
3
4
4
6
62
78
70
90
93
103
y
63.70
71.83
79.96
88.09
88.09
104.35
y–y
(y – y )2
-1.70
6.17
-9.96
1.91
4.91
-1.35
2.89
38.0689
99.2016
3.6481
24.1081
1.8225
169.7392
y  55.57  8.13 x
y  55.57  8.13 1  63.70
y  55.57  8.13  2   71.83
y  55.57  8.13  3  79.96
y  55.57  8.13  4   88.09
sest 
sest 
y  55.57  8.13  6   104.35
Bluman, Chapter 10
  y  y 
2
n2
169.7392
 6.51
4
23
Chapter 10
Correlation and Regression
Section 10-3
Example 10-13
Page #570
Bluman, Chapter 10
24
Example 10-13: Copy Machine Costs
sest 
2
y
  a y  b xy
n2
Bluman, Chapter 10
25
Example 10-13: Copy Machine Costs
sest 
sest 
Machine
Age x
(years)
A
B
C
D
E
F
1
2
3
4
4
6
Monthly
cost, y
xy
y2
62
78
70
90
93
103
62
156
210
360
372
618
3,844
6,084
4,900
8,100
8,649
10,609
496
1778
42,186
2
y
  a y  b xy
n2
42,186  55.57  496   8.13 1778
 6.48
4
Bluman, Chapter 10
26
Formula for the Prediction Interval
about a Value y 
nx  X 
1
1 
y
2
n n x 2    x 
2
y  t 2 sest
nx  X 
1
1 
2
2
n n x    x 
2
 y  t 2 sest
with d.f. = n - 2
Bluman, Chapter 10
27
Chapter 10
Correlation and Regression
Section 10-3
Example 10-14
Page #571
Bluman, Chapter 10
28
Example 10-14: Copy Machine Costs
For the data in Example 10–12, find the 95%
prediction interval for the monthly maintenance cost of
a machine that is 3 years old.
2
Step 1: Find  x,  x , and X .
 x  20  x
2
 82
20
X 
 3.3
6
Step 2: Find y for x = 3.
y  55.57  8.133  79.96
Step 3: Find sest.
sest  6.48 (as shown in Example 10-13)
Bluman, Chapter 10
29
Example 10-14: Copy Machine Costs
Step 4: Substitute in the formula and solve.
nx  X 
1
1 
y
2
2
n n x    x 
2
y  t 2 sest
nx  X 
1
1 
n n x 2    x 2
2
 y  t 2 sest
79.96   2.776  6.48  1 
6  3  3.3
2
1

y
2
6 6  82    20 
 79.96   2.776  6.48 
Bluman, Chapter 10
6  3  3.3
1
1 
6 6  82    20 2
2
30
Example 10-14: Copy Machine Costs
Step 4: Substitute in the formula and solve.
79.96   2.776  6.48 
6  3  3.3
1
1 
y
2
6 6  82    20 
 79.96   2.776  6.48 
2
6  3  3.3
1
1 
6 6  82    20 2
2
79.96  19.43  y  79.96  19.43
60.53  y  99.39
Hence, you can be 95% confident that the interval
60.53 < y < 99.39 contains the actual value of y.
Bluman, Chapter 10
31
Read section 10.3


Take notes on
Residuals.
Review the calculator
steps.


Page 574
#1-7 all, 9-17 odds
Bluman, Chapter 10
32