© 2002 Thomson / South-Western Slide 9-1 Chapter 9 Hypothesis Testing with Single Samples.

© 2002 Thomson / South-Western Slide 9-1

Chapter 9

Hypothesis Testing with

Single Samples

© 2002 Thomson / South-Western Slide 9-2

Learning ObjectivesLearning Objectives

• Understand the logic of hypothesis testing, and know how to establish null and alternate hypotheses.

• Understand Type I and Type II errors.• Use large samples to test hypotheses about

a single population mean and about a single population proportion.

• Test hypotheses about a single population mean using small samples when is unknown and the population is normally distributed.

© 2002 Thomson / South-Western Slide 9-3

Method of Indirect ProofMethod of Indirect Proof

X

X

YEither X or Y is true but not both

X is demonstrated not to be true Y

YY is true by default

© 2002 Thomson / South-Western Slide 9-4

Hypothesis Testing

A process of testing hypotheses about parameters by setting up null and alternative hypotheses, gathering sample data, computing statistics from the samples, and using statistical techniques to reach conclusions about the hypotheses.

© 2002 Thomson / South-Western Slide 9-5

Steps in Testing HypothesesSteps in Testing Hypotheses

1. Establish hypotheses: state the null and alternative hypotheses.

2. Determine the appropriate statistical test and sampling distribution.

3. Specify the Type I error rate (4. State the decision rule.5. Gather sample data.6. Calculate the value of the test statistic.7. State the statistical conclusion.8. Make a managerial decision.

© 2002 Thomson / South-Western Slide 9-6

Null and Alternative HypothesesNull and Alternative Hypotheses• The Null and Alternative Hypotheses are

mutually exclusive. Only one of them can be true.

• The Null and Alternative Hypotheses are collectively exhaustive. They are stated to include all possibilities. (An abbreviated form of the null hypothesis is often used.)

• The Null Hypothesis is assumed to be true.• The burden of proof falls on the Alternative

Hypothesis.

© 2002 Thomson / South-Western Slide 9-7

Null and Alternative Hypotheses: ExampleNull and Alternative

Hypotheses: Example• A soft drink company is filling 12 oz. cans

with cola.• The company hopes that the cans are

averaging 12 ounces.

H oz

H oz

o

a

:

:

12

12

© 2002 Thomson / South-Western Slide 9-8

Rejection and Nonrejection RegionsRejection and Nonrejection Regions

=12 oz

Nonrejection Region

Rejection Region

Critical Value

Rejection Region

Critical Value

© 2002 Thomson / South-Western Slide 9-9

Type I and Type II ErrorsType I and Type II Errors• Type I Error

– Rejecting a true null hypothesis – The probability of committing a Type I error

is called , the level of significance.

• Type II Error– Failing to reject a false null hypothesis– The probability of committing a Type II

error is called .– Power is the probability of rejecting a false

null hypothesis, and equal to 1-

© 2002 Thomson / South-Western Slide 9-10

Decision Table for Hypothesis Testing

Decision Table for Hypothesis Testing

(

( )

Null True Null False

Fail toreject null

CorrectDecision

Type II error)

Reject null Type I error

Correct Decision (Power)

© 2002 Thomson / South-Western Slide 9-11

• One-tailed Tests

One-tailed and Two-tailed TestsOne-tailed and Two-tailed Tests

H

H

o

a

:

:

12

12

H

H

o

a

:

:

12

12

H

H

o

a

:

:

12

12

• Two-tailed Test

© 2002 Thomson / South-Western Slide 9-12

One-tailed TestsOne-tailed Tests

H

H

o

a

:

:

12

12

H

H

o

a

:

:

12

12

=12 oz

Rejection Region

Nonrejection Region

Critical Value

=12 oz

Rejection Region

Nonrejection Region

Critical Value

© 2002 Thomson / South-Western Slide 9-13

Two-tailed TestsTwo-tailed Tests

H

H

o

a

:

:

12

12=12 oz

Rejection Region

Nonrejection Region

Critical Values

Rejection Region

© 2002 Thomson / South-Western Slide 9-14

CPA Net Income Example: Two-tailed Test

CPA Net Income Example: Two-tailed Test

914,74$:H

914,74$:Ho

a

If reject H .

If do not reject H .

o

o

Z Z

Z Z

c

c

196

196

. ,

. ,

78,646 74,9142.75

14,530

112

XZ

n

c oZ = 2.75 Z = 1.96, reject H

Rejection Region

Nonrejection Region

=0

Zc 196.

Rejection Region

Zc 196.

2

025.2

025.

© 2002 Thomson / South-Western Slide 9-15

CPA Net Income Example:Critical Value Method (Part 1)CPA Net Income Example:

Critical Value Method (Part 1)

Upper

nc cX Z

74 914 19614 530

11277 605

, .,

,

H

H

o

a

: $74,

: $74,

914

914

Lower

nc cX Z

74 914 19614 530

11272 223

, .,

,Rejection Region

Nonrejection Region

=0 Zc 196.

Rejection Region

Zc 196.

2

025.2

025.

72,223 77,605

© 2002 Thomson / South-Western Slide 9-16

CPA Net Income Example:Critical Value Method (Part 2)CPA Net Income Example:

Critical Value Method (Part 2)

If X or X reject H .

If 77,223 X do not reject H .

o

o

77 223 77 605

77 605

, , ,

, ,

Since X reject H .o 78 646 77 605, , ,cX

Rejection Region

Nonrejection Region

=0 Zc 196.

Rejection Region

Zc 196.

2

025.2

025.

72,223 77,605

© 2002 Thomson / South-Western Slide 9-17

Demonstration Problem 9.1 (Part 1)Demonstration Problem 9.1 (Part 1)

30.4:H

30.4:H

a

o

Rejection

Region

Nonrejection Region

0

=.05

Zc 1645.

If reject H .

If , do not reject H .

0

0

Z .

Z .

1645

1645

,Z

Xs

n

4 156 4 30

0 574

32

142. .

. .

,

do not reject H .0

Z . 142 1645.

© 2002 Thomson / South-Western Slide 9-18

Demonstration Problem 9.1 (Part 2)Demonstration Problem 9.1 (Part 2)

H

H

o

a

: .

: .

4 30

4 30

Rejection Region

Nonrejection Region

0

=.05

Zc 1645.

cx 4133. 4.30

If reject H .

If , do not reject H .

0

0

X

X

4 133

4 133

. ,

.

cX Zs

n

4 30 16450 574

324 133

. ( . ).

.

, do not reject H .0X 4 156 4 133. .

© 2002 Thomson / South-Western Slide 9-19

Rejection Region

Nonrejection Region

0

=.05

Demonstration Problem 9.1 (Part 3)Demonstration Problem 9.1 (Part 3)

H

H

o

a

: .

: .

4 30

4 30If p - value < , reject H .

If p - value , do not reject H .

o

o

Since p - value = .0778 > = .05,

do not reject H .o

Z

Xs

nP Z

4156 4 300574

32

142

142 0778

. .. .

( . ) .

© 2002 Thomson / South-Western Slide 9-20

Two-tailed Test: Small Sample, Unknown, = .05 (Part 1)

Two-tailed Test: Small Sample, Unknown, = .05 (Part 1)

Weights in Pounds of a Sample of 20 Plates

22.622.2 23.2 27.4 24.527.026.6 28.1 26.9 24.926.225.3 23.1 24.2 26.125.830.4 28.6 23.5 23.6

X 2551. , S = 2.1933, and n = 20

© 2002 Thomson / South-Western Slide 9-21

Two-tailed Test: Small Sample, Unknown, = .05 (Part 2)

Two-tailed Test: Small Sample, Unknown, = .05 (Part 2)

Critical Values

Nonrejection Region

Rejection Regions

ct 2 093. ct 2 093.

2

025.2

025.

H

H

o

a

:

:

25

25

df n 1 19

© 2002 Thomson / South-Western Slide 9-22

Two-tailed Test: Small Sample, Unknown, = .05 (Part 3)

Two-tailed Test: Small Sample, Unknown, = .05 (Part 3)

tX

S

n

2551 25 0

2 1933

20

104. .

. .

Since t do not reject H .o 104 2 093. . ,Critical Values

Non Rejection Region

Rejection Regions

ct 2 093. ct 2 093.

2

025.2

025.

If t reject H .

If t do not reject H .

o

o

2 093

2 093

. ,

. ,

© 2002 Thomson / South-Western Slide 9-23

Demonstration Problem 9.2 (Part 1)Demonstration Problem 9.2 (Part 1)

Size in Acres of 23 Farms

445 489474505 553 477 545463 466557502 449 438 500466 477557433 545 511 590561 560

23 = and 46.94,= ,78.498 nSX

© 2002 Thomson / South-Western Slide 9-24

Demonstration Problem 9.2 (Part 2)Demonstration Problem 9.2 (Part 2)

471:

471:

a

o

H

H

df n 1 22

Critical Value

Nonrejection Region

Rejection Region

ct 1717.

.05

© 2002 Thomson / South-Western Slide 9-25

Demonstration Problem 9.2 (Part 3)Demonstration Problem 9.2 (Part 3)

If t reject H .

If t do not reject H .

o

o

1717

1717

. ,

. ,

84.2

23

94.4647178.498

n

SX

t

.Hreject ,717.184.2 tSince oCritical Value

Nonrejection Region

Rejection Region

ct 1717.

.05

© 2002 Thomson / South-Western Slide 9-26

Z Test of Population ProportionZ Test of Population Proportion

Zp P

P Qn

where

: p = sample proportion

P = population proportion

Q = 1 - P

n P

n Q

5

5

, and

© 2002 Thomson / South-Western Slide 9-27

Testing Hypotheses about a Proportion: Manufacturer Example

(Part 1)

Testing Hypotheses about a Proportion: Manufacturer Example

(Part 1)

08.:H

08.:H

a

o

P

P

cZ 1645.

Critical Values

Nonrejection Region

Rejection Regions

cZ 1645.

2

05. 2

05.

© 2002 Thomson / South-Western Slide 9-28

Testing Hypotheses about a Proportion: Manufacturer Example

(Part 2)

Testing Hypotheses about a Proportion: Manufacturer Example

(Part 2)

.

. .

(. )(. ).

p

Zp P

P Qn

33

200165

165 08

08 92200

4 43

If Z reject H .

If Z do not reject H .

o

o

1645

1645

. ,

. ,

Since Z reject H .o 4 43 1645. . ,

cZ 1645.

Critical Values

Nonrejection Region

Rejection Regions

cZ 1645.

2

05. 2

05.

© 2002 Thomson / South-Western Slide 9-29

Demonstration Problem 9.3 (Part 1)Demonstration Problem 9.3 (Part 1)

H P

H P

o

a

: .

: .

17

17

Critical Value

Nonrejection Region

Rejection Region

cZ 1645.

.05

© 2002 Thomson / South-Western Slide 9-30

Demonstration Problem 9.3 (Part 2)Demonstration Problem 9.3 (Part 2)

.

. .

(. )(. ).

p

Zp P

P Qn

115

550209

209 17

17 83550

2 44

If reject H .

If do not reject H .

o

o

Z

Z

1645

1645

. ,

. ,

Since Z = 2.44 reject H .o1645. ,Critical Value

Nonrejection Region

Rejection Region

cZ 1645.

.05