10.3 coefficient of determination and standard error of the ......residuals the values of...
TRANSCRIPT
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
Bluman, Chapter 10 5
Regression
line
Variations
The total variation is
calculated by:
This the sum of the squares of the
vertical distances from the mean.
(𝑦 − 𝑦 )2
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
Most of the variation can be
explained by the relationship.
(𝑦′ − 𝑦 )2
Unexplained variation
2')( yy
When this variation is small then the
value of r will be close to 1 or -1.
The last few slides summarized:
Explained variation Unexplained variation Total variation
(𝑦 − 𝑦 )2= (𝑦′ − 𝑦 )2+ (𝑦 − 𝑦′)2
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, r2
2.non-determination,
1-r2
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.
Another way to arrive at the value for r 2
is to square the correlation coefficient.
Bluman, Chapter 10 14
2 explained variation
total variationr
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.
i.e. if r2=.81 that means 81% of variation
in the dependent variable is explained by
the variation in the independent variable.
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.
2
2
n
xybyayS
est
2
)( 2'
n
yys
est
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:
Bluman, Chapter 10 20
2
2
est
y ys
n
Chapter 10 Correlation and Regression
Section 10-3
Example 10-12
Page #569
Bluman, Chapter 10 21
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.
Example 10-12: Copy Machine Costs
Bluman, Chapter 10 22
Example 10-12: Copy Machine Costs
Bluman, Chapter 10 23
Machine
Age x
(years)
Monthly
cost, y y y – y (y – y ) 2
A
B
C
D
E F
1
2
3
4
4 6
62
78
70
90
93 103
63.70
71.83
79.96
88.09
88.09 104.35
- 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
55.57 8.13
55.57 8.13 1 63.70
55.57 8.13 2 71.83
55.57 8.13 3 79.96
55.57 8.13 4 88.09
55.57 8.13 6 104.35
y x
y
y
y
y
y
2
2
est
y ys
n
169.73926.51
4 ests
Chapter 10 Correlation and Regression
Section 10-3
Example 10-13
Page #570
Bluman, Chapter 10 24
Example 10-13: Copy Machine Costs
Bluman, Chapter 10 25
2
2
est
y a y b xys
n
Example 10-13: Copy Machine Costs
Bluman, Chapter 10 26
496 1778
Machine
Age x
(years)
Monthly
cost, y xy y2
A
B
C
D
E F
1
2
3
4
4 6
62
78
70
90
93 103
62
156
210
360
372 618
3,844
6,084
4,900
8,100
8,649 10,609
2
2
est
y a y b xys
n
42,186
42,186 55.57 496 8.13 17786.48
4
ests
Formula for the Prediction Interval
about a Value y
Bluman, Chapter 10 27
2
2 22
2
2 22
11
11
with d.f. = - 2
est
est
n x Xy t s y
n n x x
n x Xy t s
n n x x
n
Chapter 10 Correlation and Regression
Section 10-3
Example 10-14
Page #571
Bluman, Chapter 10 28
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.
Step 1: Find
Step 2: Find y for x = 3.
Step 3: Find sest.
(as shown in Example 10-13)
Example 10-14: Copy Machine Costs
Bluman, Chapter 10 29
2, , and . x x X
2 2020 82 3.3
6 x x X
55.57 8.13 3 79.96 y
6.48ests
Step 4: Substitute in the formula and solve.
Example 10-14: Copy Machine Costs
Bluman, Chapter 10 30
2
2 22
2
2 22
11
11
est
est
n x Xy t s y
n n x x
n x Xy t s
n n x x
2
2
2
2
6 3 3.3179.96 2.776 6.48 1
6 6 82 20
6 3 3.3179.96 2.776 6.48 1
6 6 82 20
y
Step 4: Substitute in the formula and solve.
Example 10-14: Copy Machine Costs
Bluman, Chapter 10 31
2
2
2
2
6 3 3.3179.96 2.776 6.48 1
6 6 82 20
6 3 3.3179.96 2.776 6.48 1
6 6 82 20
y
79.96 19.43 79.96 19.43
60.53 99.39
y
y
Hence, you can be 95% confident that the interval
60.53 < y < 99.39 contains the actual value of y.
Read section 10.3
Take notes on
Residuals.
Review the calculator
steps.
Page 574
#1-7 all, 9-17 odds
Bluman, Chapter 10 32