商務(wù)與經(jīng)濟統(tǒng)計ppt第11章課件

Statistics

for

Businessand

Economics?

2011

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide1Anderson

Sweeney

WilliamsSlides

byJohn

LoucksSt.

Edward’s

UniversityChapter

11Inferences

About

Population

VariancesInference

about

a

Population

VarianceInferences

about

Two

Populations

Variances?

2011

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website,

in

whole

or

inpart.Slide2Inferences

About

a

Population

VarianceA

variancecan

provide

importantdecision-makinginformation.Consider

the

production

process

of

fillingcontainerswith

a

liquid

detergent

product.The

mean

filling

weight

is

important,

but

also

is

the variance

of

the

fillingweights.By

selecting

a

sample

of

containers,

we

cancompute a

sample

variance

for

the

amount

of

detergentplaced in

a

container.If

the

sample

variance

is

excessive,

overfilling

and underfilling

may

be

occurring

even

though

the

mean is

correct.?

2011

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Learning.

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Reserved.

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide3Inferences

About

a

Population

Variance?

2011

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to

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accessible

website,

in

whole

or

inpart.Slide42Chi-SquareDistributionInterval

Estimation

of

Hypothesis

TestingChi-SquareDistributionThe

chi-square

distribution

is

the

sum

ofsquared standardized

normal

random

variables

such

as (z1)2+(z2)2+(z3)2

and

so

on.Thechi-square

distribution

is

based

on

sampling from

a

normalpopulation.The

sampling

distribution

of

(n

-

1)s2/

2

has

a

chi- square

distribution

whenever

a

simple

random

sample of

sizen

is

selected

from

a

normalpopulation.We

can

use

thechi-square

distribution

todevelop interval

estimates

and

conduct

hypothesis

tests about

a

population

variance.?

2011

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posted

to

a

publicly

accessible

website,

in

whole

or

inpart.Slide5Examples

of

Sampling

Distribution

of

(n

-

1)s2/

20With

2

degreesoffreedomWith

5

degreesoffreedomWith

10

degreesoffreedom?

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to

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accessible

website,

in

whole

or

inpart.Slide6Chi-Square

DistributionWewillusethe

notationtodenotethe

value

forthechi-square

distributionthatprovidesanareaof

α

tothe

right

ofthestatedvalue.For

example,

there

is

a

.95

probability

of

obtaining

a

2(chi-square)value

such

that?

2011

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website,

in

whole

or

inpart.Slide7

20.02595%ofthepossible

2

values.025Interval

Estimation

of

?

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide82Interval

Estimation

of

2Performing

algebraic

manipulation

wegetThere

is

a

(1

–

α)

probability

of

obtaining

a

2

value such

thatSubstituting(n–

1)s2/σ

2

forthe

2

weget?

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duplicated,

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide9Interval

Estimate

of

a

Population

VarianceInterval

Estimation

of

2where

the

values

are

basedona

chi-squaredistribution

withn

-

1

degrees

of

freedom

andwhere

1

-

is

the

confidencecoefficient.?

2011

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Learning.

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Rights

Reserved.

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scanned,

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duplicated,

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posted

to

a

publicly

accessible

website,

in

whole

or

inpart.Slide10Interval

Estimation

of

Interval

Estimate

of

a

Population

Standard

DeviationTaking

the

square

root

of

the

upper

and

lowerlimits

of

the

variance

interval

provides

the

confidenceinterval

for

the

population

standarddeviation.?

2011

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Learning.

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Rights

Reserved.

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not

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scanned,

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duplicated,

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posted

to

a

publicly

accessible

website,

in

whole

or

inpart.Slide11Interval

Estimation

of

2Example: Buyer’s

Digest

(A)Buyer’s

Digest

rates

thermostats

manufactured

forhome

temperature

control. In

a

recent

test,

10thermostats

manufactured

by

ThermoRite

wereselected

and

placed

in

a

test

room

that

wasmaintained

at

a

temperature

of

68oF.

Thetemperature

readings

of

the

ten

thermostatsareshown

on

the

nextslide.?

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accessible

website,

in

whole

or

inpart.Slide12Interval

Estimation

of

2Example:

Buyer’s

Digest

(A)We

will

use

the

10

readings

below

to

develop

a95%

confidence

interval

estimate

of

the

populationvariance.ThermostatTemperature1

2

3

4

5

6

7

8

9

1067.4

67.8

68.2

69.3

69.5

67.0

68.1

68.6

67.9

67.2?

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accessible

website,

in

whole

or

inpart.Slide13Interval

Estimation

of

2Selected

Values

from

the

Chi-Square

Distribution

TableOurvalueFor

n-1=

10-

1=

9d.f.and

α

=

.05?

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website,

in

whole

or

inpart.Slide14Interval

Estimation

of

2

2.025Area

inUpperTail=

.9750

2.700For

n-1=

10-

1=

9d.f.and

α

=

.05?

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide15Interval

Estimation

of

2Selected

Values

from

the

Chi-Square

Distribution

TableFor

n-1=

10-

1=

9d.f.and

α

=

.05Ourvalue?

2011

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Learning.

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Rights

Reserved.

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posted

to

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website,

in

whole

or

inpart.Slide16

2.0250

2.700Interval

Estimation

of

2n

-1

=

10-

1=

9

degrees

offreedom

and

α

=

.0519.023?

2011

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website,

in

whole

or

inpart.Slide17Area

inUpperTail

=

.025Sample

variance

s2

provides

a

point

estimate

of

2.Interval

Estimation

of

2.33<

2

<

2.33A

95%

confidence

interval

for

the

population

variance

isgiven

by:?

2011

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Reserved.

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scanned,

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide18Hypothesis

TestingAbout

a

Population

Variancewhere is

the

hypothesizedvaluefor

the

population

varianceTest

Statistic?

2011

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Reserved.

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copiedor

duplicated,

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posted

to

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accessible

website,

in

whole

or

inpart.Slide19Left-TailedTestHypothesesLeft-Tailed

Test

(continued)Hypothesis

TestingAbout

a

Population

VarianceReject

H0

if

p-value

<

αp-Value

approach:Rejection

RuleCritical

value

approach:Reject

H0

ifwhere is

based

onachi-squaredistribution

withn

-

1

d.f.?

2011

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to

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website,

in

whole

or

inpart.Slide20Hypothesis

TestingAbout

a

Population

Variancewhere is

the

hypothesizedvaluefor

the

population

varianceTest

Statistic?

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Reserved.

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to

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accessible

website,

in

whole

or

inpart.Slide21Right-Tailed

TestHypothesesRight-TailedTest(continued)Hypothesis

TestingAbout

a

Population

VarianceReject

H0

ifReject

H0

if

p-value

<

αwhere is

based

onachi-squaredistribution

withn

-

1

d.f.p-Value

approach:?

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Reserved.

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to

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accessible

website,

in

whole

or

inpart.Slide22Rejection

RuleCritical

value

approach:Hypothesis

TestingAbout

a

Population

Variancewhere is

the

hypothesizedvaluefor

the

population

varianceTest

Statistic?

2011

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Learning.

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Rights

Reserved.

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scanned,

copiedor

duplicated,

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posted

to

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publicly

accessible

website,

in

whole

or

inpart.Slide23Two-Tailed

TestHypothesesHypothesis

TestingAbout

a

Population

VarianceReject

H0

if

p-value

<

αp-Value

approach:Two-Tailed

Test

(continued)Rejection

RuleCritical

value

approach:Reject

H0

ifwhere are

based

on

achi-square

distribution

with

n

-

1d.f.?

2011

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Rights

Reserved.

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not

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scanned,

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duplicated,

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posted

to

a

publicly

accessible

website,

in

whole

or

inpart.Slide24Hypothesis

TestingAbout

a

Population

VarianceExample: Buyer’s

Digest

(B)Recall

that

Buyer’s

Digest

is

rating

ThermoRitethermostats.

Buyer’s

Digest

gives

an

“acceptable”rating

to

a

thermostat

with

a

temperature

varianceof

0.5

orless.We

will

conduct

a

hypothesis

test

(with

α

=

.10)to

determine

whether

the

ThermoRite

thermostat’stemperature

variance

is

“acceptable”.?

2011

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Reserved.

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scanned,

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to

a

publicly

accessible

website,

in

whole

or

inpart.Slide25Hypothesis

TestingAbout

a

Population

VarianceExample: Buyer’s

Digest

(B)Using

the

10

readings,

we

will

conduct

ahypothesis

test

(with

α

=

.10)

to

determinewhetherthe

ThermoRite

thermostat’s

temperature

variance

is“acceptable”.Thermostat

1

2

3

4

5

6

7

8

9

10Temperature

67.4

67.8

68.2

69.3

69.5

67.0

68.1

68.6

67.9

67.2?

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to

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publicly

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website,

in

whole

or

inpart.Slide26HypothesesHypothesis

TestingAbout

a

Population

VarianceReject

H0

if

χ

2

>

14.684Rejection

RuleRight-tailedtest?

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to

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accessible

website,

in

whole

or

inpart.Slide27Selected

Values

from

the

Chi-Square

Distribution

TableFor

n-1=

10-

1=

9d.f.and

α

=

.10Hypothesis

TestingAbout

a

Population

VarianceOurvalue?

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website,

in

whole

or

inpart.Slide28014.684Area

inUpperTail

=

.10

2Hypothesis

TestingAbout

a

Population

VarianceRejection

RegionReject

H0?

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website,

in

whole

or

inpart.Slide29Hypothesis

TestingAbout

a

Population

VarianceConclusionBecause

2

=

12.6

is

less

than

14.684,

we

cannotreject

H0. The

sample

variance

s2

=

.7

isinsufficientevidence

to

conclude

that

the

temperature

variancefor

ThermoRite

thermostats

is

unacceptable.Test

StatisticThe

sample

variance

s

2

=

0.7?

2011

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide30Hypothesis

TestingAbout

a

Population

VarianceUsing

the

p-ValueThe

rejection

region

for

theThermoRite thermostat

example

is

in

the

upper

tail;

thus,

the appropriate

p-value

is

lessthan

.90(χ

2

=

4.168) and

greaterthan.10(χ

2

=

14.684).Because

the

p

–value

>α

=

.10,

we

cannot reject

the

nullhypothesis.The

sample

variance

of

s2

=

.7

is

insufficient evidence

to

conclude

that

the

temperature variance

is

unacceptable

(>.5).The

exact

p-value

is.18156.?

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to

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publicly

accessible

website,

in

whole

or

inpart.Slide31Inferences

About

Two

Population

VariancesWe

may

want

to

compare

the

variances

in:product

quality

resulting

from

twodifferentproduction

processes,temperatures

for

two

heating

devices,

orassembly

times

for

two

assembly

methods.We

use

data

collected

from

two

independent

randomsample,

one

from

population

1

and

another

frompopulation

2.The

two

sample

variances

will

be

the

basis

formaking inferences

about

the

two

populationvariances.?

2011

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to

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publicly

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website,

in

whole

or

inpart.Slide32One-TailedTestHypothesesHypothesis

Testing

About

theVariances

of

TwoPopulationsDenote

the

population

providing

thelarger

sample

variance

as

population

1.Test

Statistic?

2011

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to

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accessible

website,

in

whole

or

inpart.Slide33One-Tailed

Test(continued)Reject

H0

if

p-value

<

αwhere

the

value

ofF

is

based

on

anF

distribution

with

n1

-

1

(numerator)and

n2-1

(denominator)d.f.p-Value

approach:Rejection

RuleCritical

value

approach:Hypothesis

Testing

About

theVariances

of

TwoPopulationsReject

H0

if

F>F

?

2011

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website,

in

whole

or

inpart.Slide34Two-Tailed

TestHypothesesHypothesis

Testing

About

theVariances

of

TwoPopulationsDenote

the

population

providing

thelarger

sample

variance

as

population

1.Test

Statistic?

2011

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Reserved.

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to

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accessible

website,

in

whole

or

inpart.Slide35Two-Tailed

Test

(continued)Reject

H0

if

p-value

<

αp-Value

approach:?

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website,

in

whole

or

inpart.Slide36Rejection

RuleCritical

value

approach:Hypothesis

Testing

About

theVariances

of

TwoPopulationsReject

H0

if

F>F

/2where

the

value

ofF

/2

is

based

on

anF

distribution

with

n1

-

1

(numerator)and

n2-1

(denominator)d.f.Hypothesis

Testing

About

theVariances

of

TwoPopulationsExample: Buyer’s

Digest

(C)Buyer’s

Digest

has

conducted

the

same

test,

as

wasdescribed

earlier,

on

another

10

thermostats,

this

timemanufactured

by

TempKing. The

temperature

readingsof

the

ten

thermostats

are

listed

on

the

next

slide.We

will

conduct

a

hypothesis

test

with

=

.10

to

seeif

the

variances

are

equal

for

ThermoRite’sthermostatsand

TempKing’sthermostats.?

2011

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Reserved.

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to

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website,

in

whole

or

inpart.Slide37Hypothesis

Testing

About

theVariances

of

TwoPopulationsExample: Buyer’s

Digest

(C)ThermoRite

SampleThermostat

1

2

3

4

5

6

7

8

9

10Temperature

67.4

67.8

68.2

69.3

69.5

67.0

68.1

68.6

67.9

67.2TempKing

SampleThermostat

1

2

3

4

5

6

7

8

9

10Temperature

67.7

66.4

69.2

70.1

69.5

69.7

68.1

66.6

67.3

67.5?

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Reserved.

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scanned,

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to

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publicly

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website,

in

whole

or

inpart.Slide38Hypothesis

Testing

About

theVariances

of

TwoPopulationsReject

H0

ifF

>3.18Hypotheses(TempKing

and

ThermoRite

thermostatshave

the

same

temperature

variance)(Their

variances

are

not

equal)Rejection

RuleThe

F

distribution

table

(on

next

slide)

shows

that

withwith

=

.10,

9

d.f.(numerator),

and

9d.f.

(denominator),F.05

=

3.18.?

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