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