Sec 10.3
description
Transcript of Sec 10.3
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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
′
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
′ 7.6 10.4 13.2 16 18.8
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Test 10-11Friday April 18
Monday April 7Periods 5-8 assembly schedule
MONDAY 21 PERIODS 5-8 WED SCHEDULE.
Wed April 23 ACTOur class doesn’t meet
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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.
2)(
yy
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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
2
2'
)(
)(
yy
yy2' )(
yy
Most of the variation can be explained by the relationship.
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Unexplained variation
2')( yyWhen 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:
2'2'2 )()()( yyyyyy
Explained variation Unexplained variationTotal variation
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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 variationtotal variation
r
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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 – 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% …
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Example:
Let r=0.9123Find 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
xybyaySest
2)( 2'
n
yysest
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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 10Correlation and Regression
Section 10-3Example 10-12Page #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
MachineAge x
(years)Monthlycost, y y y – y (y – y )2
ABCDEF
123446
6278709093103
63.7071.8379.9688.0988.09
104.35
-1.706.17
-9.961.914.91
-1.35
2.8938.068999.2016
3.648124.1081
1.8225
169.7392
55.57 8.1355.57 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 xy
y
y
y
y
2
2
est
y ys
n
169.7392 6.514
ests
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Chapter 10Correlation and Regression
Section 10-3Example 10-13Page #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
MachineAge x
(years)Monthlycost, y xy y2
ABCDEF
123446
6278709093
103
62156210360372618
3,8446,0844,9008,1008,649
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 10Correlation and Regression
Section 10-3Example 10-14Page #571
Bluman, Chapter 10 28
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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 X2 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 16 6 82 20
6 3 3.3179.96 2.776 6.48 16 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 16 6 82 20
6 3 3.3179.96 2.776 6.48 16 6 82 20
y
79.96 19.43 79.96 19.4360.53 99.39
yy
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