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Sec 10.3
Coefficient of Determination and Standard
Error of the Estimate.
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Review concepts
x 1 2 3 4 5
y 10 8 12 16 20
πβ²
fill in the third row of the table for each x
value.
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Review concepts
x 1 2 3 4 5
y 10 8 12 16 20
πβ² 7.6 10.4 13.2 16 18.8
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Variation
Bluman, Chapter 10 5
Regression
line
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Variations
The total variation is
calculated by:
This the sum of the squares of the
vertical distances from the mean.
(π¦ β π¦ )2
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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.
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Explained variation
Most of the variation can be
explained by the relationship.
(π¦β² β π¦ )2
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Unexplained variation
2')( yy
When this variation is small then the
value of r will be close to 1 or -1.
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The last few slides summarized:
Explained variation Unexplained variation Total variation
(π¦ β π¦ )2= (π¦β² β π¦ )2+ (π¦ β π¦β²)2
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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.
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Coefficient of . . .
1.Determination, r2
2.non-determination,
1-r2
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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.
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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
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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.
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Coefficient of Nondetermiation
The coefficient of nondetermination is
a measure of the unexplained variation.
The formula for the coefficient of
determination is 1.00 β r 2.
Bluman, Chapter 10 16
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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% β¦
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Example:
Let r=0.9123
Find the coefficients of
determination and
nondetermination.
Explain the meaning of each.
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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
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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
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Chapter 10 Correlation and Regression
Section 10-3
Example 10-12
Page #569
Bluman, Chapter 10 21
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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
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Example 10-12: Copy Machine Costs
Bluman, Chapter 10 23
Machine
Age x
(years)
Monthly
cost, y y y β y (y β 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
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Chapter 10 Correlation and Regression
Section 10-3
Example 10-13
Page #570
Bluman, Chapter 10 24
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Example 10-13: Copy Machine Costs
Bluman, Chapter 10 25
2
2
est
y a y b xys
n
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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
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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
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Chapter 10 Correlation and Regression
Section 10-3
Example 10-14
Page #571
Bluman, Chapter 10 28
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For the data in Example 10β12, 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
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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
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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.
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Read section 10.3
Take notes on
Residuals.
Review the calculator
steps.
Page 574
#1-7 all, 9-17 odds
Bluman, Chapter 10 32