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Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.
32

Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Dec 24, 2015

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Page 1: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Sec 10.3

Coefficient of Determination and Standard Error of the Estimate.

Page 2: 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

fill in the third row of the table for each x value.

Page 3: 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

′ 7.6 10.4 13.2 16 18.8

Page 4: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 5: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Variation

Bluman, Chapter 10 5

Regression line

Page 6: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Variations

The total variation is calculated by:

This the sum of the squares of the vertical distances from the mean.

2)(

yy

Page 7: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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.

Page 8: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Explained variation

2

2'

)(

)(

yy

yy

2' )(

yy

Most of the variation can be explained by the relationship.

Page 9: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Unexplained variation

2')( yy

When this variation is small then the value of r will be close to 1 or -1.

Page 10: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

The last few slides summarized:

2'2'2 )()()( yyyyyy

Explained variation Unexplained variationTotal variation

Page 11: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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.

Page 12: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Coefficient of . . .

1.Determination, r2

2.non-determination, 1-r2

Page 13: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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.

Page 14: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 15: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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.

Page 16: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 17: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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% …

Page 18: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Example:

Let r=0.9123Find the coefficients of

determination and nondetermination.

Explain the meaning of each.

Page 19: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

yys

est

Page 20: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 21: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Chapter 10Correlation and Regression

Section 10-3Example 10-12

Page #569

Bluman, Chapter 10 21

Page 22: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 23: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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 8.13

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

Page 24: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Chapter 10Correlation and Regression

Section 10-3Example 10-13

Page #570

Bluman, Chapter 10 24

Page 25: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Example 10-13: Copy Machine Costs

Bluman, Chapter 10 25

2

2

est

y a y b xys

n

Page 26: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 27: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 28: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Chapter 10Correlation and Regression

Section 10-3Example 10-14

Page #571

Bluman, Chapter 10 28

Page 29: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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 20

20 82 3.36

x x X

55.57 8.13 3 79.96 y

6.48ests

Page 30: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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

Page 31: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

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.

Page 32: Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

Read section 10.3

Take notes on Residuals.

Review the calculator steps.

Page 574 #1-7 all, 9-17 odds

Bluman, Chapter 10 32