hypothesis testing slides

8/7/2019 hypothesis testing slides

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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Chapter 8

Tests ofHypotheses Based

on a SingleSample


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8.1

Hypothesesand Test

Procedures


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Hypotheses

The null hypothesis, denotedH0, is the

claim that is initially assumed to be true.

The alternative hypothesis, denoted byHa, is the assertion that is contrary toH0.

Possible conclusions from hypothesis-

testing analysis are reject H0 orfail toreject H0.


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Hypotheses

H0 may usually be considered the

skeptics hypothesis: Nothing new or

interesting happening here! (Andanything interesting observed is due to

chance alone.)

Ha may usually be considered the

researchers hypothesis.


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Rules for Hypotheses

H0 is always stated as an equality claim

involving parameters.

Ha is an inequality claim that contradicts

H0. It may be one-sided (using either >

or


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A Test of Hypotheses

A test of hypotheses is a method forusing sample data to decide whether

the null hypothesis should be

rejected.


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Test Procedure

A test procedure is specified by1. A test statistic, a function of the

sample data on which the decision is

to be based.

2. (Sometimes, not always!) A

rejection region, the set of all test

statistic values for whichH0 will be

rejected


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Errors in Hypothesis Testing

A type I errorconsists of rejecting the

null hypothesisH0 when it was true.A type II errorconsists of not rejecting

H0 whenH0 is false.

andE F are the probabilities of typeI and type II error, respectively.


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Level TestE

A test corresponding to the significance

level is called a level test. A test

with significance level is one forwhich the type I error probability is

controlled at the specified level.

E

E

Sometimes, the experimenter will fix

the value of , also known as the

significance level.

E


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Rejection Region: andE F

Suppose an experiment and a sample

size are fixed, and a test statistic is

chosen. Decreasing the size of the

rejection region to obtain a smaller

value of results in a larger value of

for any particular parameter value

consistent withHa.

E F


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8.2

Tests Abouta

Population Mean


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Case I: A Normal Population

With Known W

Null hypothesis: 0 0:H Q Q!

Test statistic value:0

/

x

zn

Q

W


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Case I: A Normal Population

With Known W

a 0:H Q Q"

Alternative

HypothesisRejection Region

for Level Test

a 0:H Q Qa 0:H Q Q{

Ez zEu

z zEe

/ 2z zEu / 2z zEe or


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Recommended Steps in

Hypothesis-Testing Analysis

1. Identify the parameter of interest and

describe it in the context of theproblem situation.

2. Determine the null value and state

the null hypothesis.

3. State the alternative hypothesis.


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4. Give the formula for the computed

value of the test statistic.

5. State the rejection region for theselected significance level

6. Compute any necessary sample

quantities, substitute into the formula

for the test statistic value, and

compute that value.


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7. Decide whetherH0 should be

rejected and state this conclusion in

the problem context.

The formulation of hypotheses (steps 2

and 3) should be done before examiningthe data.


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Type II Probability for a Level

Test

E( )F Qd

Alt. Hypothesis

a 0

:H Q Q"

a 0:H Q Q

a 0:H Q Q{

Type IIProbability ( )F Qd

0

/z

nE

Q Q

W

d *

01/

z

nE

Q Qd *

0 0/ 2 / 2

/ /z z

n nE E

Q Q Q Qd d * *


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Sample Size

The sample size n for which a level

test also has at the alternative

value is

( )F Q Fd!Qd

E

2

0

2

/ 2

0

( )

( )

z z

nz z

E F

E F

W

Q Q

W

Q Q

d

!

d

one-tailed test

two-tailed test


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Case II: Large-Sample Tests

When the sample size is large, the z

tests for case I are modified to yieldvalid test procedures without

requiring either a normal population

distribution or a known .W


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Large Sample Tests (n > 40)

For large n, s is close to .W

Test Statistic: 0/

XZ

S nQ!

The use of rejection regions for case I

results in a test procedure for which the

significance level is approximately .E


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Case III: A Normal Population

Distribution

IfX1,,Xn is a random sample from a

normal distribution, the standardizedvariable

has a tdistribution with n 1

degrees of freedom.

/

XT

S

n

Q!


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The One-Sample tTest

Null hypothesis:0 0:H Q Q!

Test statistic value: 0

/

x

ts n

Q!


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a 0:H Q Q"

Alternative

Hypothesis

Rejection Region

for Level Test

a 0:H Q Qa 0:H Q Q{

E, 1nt tE u

, 1nt tE e

/ 2, 1nt tE or

The One-Sample tTest

/ 2, 1nt tE e


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8.3

Tests Concerninga

Population Proportion


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A Population Proportion

Letp denote the proportion of

individuals or objects in a

population who possess a specified

property.


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Large-Sample Tests

Large-sample tests concerningpare a special case of the more

general large-sample procedures

for a parameter.U


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Large-Samples Concerningp

Null hypothesis:0 0:H p p!

Test statistic value:

0

0 0

1 /

p pz

p p n

!


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a 0

:H p p"

AlternativeHypothesis

Rejection Region

a 0:H p p

a 0:H p p{

z z

E

u

z zEe

/ 2z zEu / 2z zEe or

Large-Samples Concerningp

Valid provided

0 010 and (1 ) 10.np n pu u


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( )pF d

Alt. Hypothesis

a 0:H p p"

a 0:H p p

( )pF d

0 0 0(1 ) /

(1 ) /

p p z p p n

p p n

E d

* d d

General Expressions for

0 0 0(1 ) /1

(1 ) /

p p z p p n

p p n

E d

* d d


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( )pF d

Alt. Hypothesis

a 0:H p p{

( )pF d

General Expressions for

0 0 0(1 ) /

(1 ) /

p p z p p n

p p n

E d * d d

0 0 0(1 ) /

(1 ) /

p p z p p n

p p n

E d * d d


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Sample Size

The sample size n for which a level

test also has ( )p pF d!E

20 0

0

2

/ 2 0 0

0

(1 ) (1 )

(1 ) (1 )

z p p z p p

p pn

z p p z p pp p

E F

E F

d d d

!

d d d

two-tailedtest

one-tailedtest


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Small-Sample Tests

Test procedures when the sample size

n is small are based directly on the

binomial distribution rather than thenormal approximation.

0(type I) 1 ( 1; , )P B c n p! ( ) ( 1; , )B p B c n pd d!


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8.4

P- Values


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P- Value

TheP-value is the smallest level of

significance at whichH0 would be

rejected when a specified test procedure

is used on a given data set.

0

1. -value

reject at a level of

P

H

E

E

e

0

2. -value

do not reject at a level of

P

H

E

E

"


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P- Value

TheP-value is the probability,

calculated assumingH0 is true, of

obtaining a test statistic value at least ascontradictory toH0 as the value that

actually resulted. The smaller theP-

value, the more contradictory is the datatoH0.


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P-Values for a z Test

P-value:

1 ( )

( )

2 1 ( )

z

P z

z

*

! *

*

upper-tailed test

lower-tailed test

two-tailed test


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P-Value (area)

z

-z

-value 1 ( )P z! *

-value ( )P z! *

-value 2[1 (| |)]P z! *

0

0

0

-z

z

Upper-Tailed

Lower-Tailed

Two-Tailed


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PValues fortTests

TheP-value for a ttest will be a tcurve area. The number of df for the

one-sample ttest is n 1.


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8.5

Some Comments onSelecting a

Test Procedure


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Constructing a Test Procedure

1. Specify a test statistic.

2. Decide on the general form of the

rejection region.

3. Select the specific numerical critical

value or values that will separate the

rejection region from the acceptance

region.


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Issues to be Considered

1. What are the practical implications

and consequences of choosing a

particular level of significance once

the other aspects of a test procedure

have been determined?

2. Does there exist a general principlethat can be used to obtain best or good

test procedures?


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Issues to be Considered

3. When there exist two or more tests thatare appropriate in a given situation, how

can the tests be compared to decide

which should be used?

4. If a test is derived under specific

assumptions about the distribution of

the population being sampled, how well

will the test procedure work when the

assumptions are violated?


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StatisticalVersus Practical

Significance

Be careful in interpreting evidence when

the sample size is large, since any smalldeparture fromH0 will almost surely be

detected by a test (statistical significance),

yet such a departure may have little

practical significance.


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The Likelihood Ratio Principle

1. Find the largest value of the likelihoodfor any

2. Find the largest value of the likelihoodfor any

3. Form the ratio

0in .U ;

ain .U ;

01a

maximum likelihood for in,...,maximum likelihood for in

nx xUPU

;!;

RejectH0 when this ratio is small.

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