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

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.

Review concepts

x 1 2 3 4 5

y 10 8 12 16 20

′ 7.6 10.4 13.2 16 18.8

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

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.

2)(

yy

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

2

2'

)(

)(

yy

yy

2' )(

yy

Most of the variation can be explained by the relationship.

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:

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

Explained variation Unexplained variationTotal variation

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.


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.


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=0.9123Find 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

xybyaySest

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:


2

2

est

y ys

n

Chapter 10Correlation and Regression

Section 10-3Example 10-12

Page #569


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




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 #570




2

2

est

y a y b xys

n



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

Formula for the Prediction Interval about a Value y


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 #571


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)



2, , and . x x X2 20

20 82 3.36

x x X

55.57 8.13 3 79.96 y

6.48ests

Step 4: Substitute in the formula and solve.



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.



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


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

Documents

variation bluman

coefficient nondetermination

parts of variation

types of variation

correlation coefficient

value of r

regression line slide

values of y