© 2002 Thomson / South-Western Slide 10-1

Chapter 10

HypothesisTesting with Two Samples

© 2002 Thomson / South-Western Slide 10-2

Learning ObjectivesLearning Objectives

• Test hypotheses about the difference in two population means using data from large independent samples.

• Test hypotheses about the difference in two population means using data from small independent samples when the populations are normally distributed.

© 2002 Thomson / South-Western Slide 10-3

Learning Objectives, continued

• Test hypotheses about the mean difference in two related populations when the populations are normally distributed.

• Test hypotheses about the differences in two population proportions.

• Test hypotheses about two population variances when the populations are normally distributed.

© 2002 Thomson / South-Western Slide 10-4

Hypothesis Testing about the Difference in Two Sample Means

Hypothesis Testing about the Difference in Two Sample Means

11

X nx

Population 1

Population 2

22

X nx

1 2X X

1X

2X

1 2X X

© 2002 Thomson / South-Western Slide 10-5

Hypothesis Testing about the Difference in Two Sample Means

Hypothesis Testing about the Difference in Two Sample Means

1 2X X1 2X X

1 2

1

2

1

2

2

2X X n n

1 21 2X X

© 2002 Thomson / South-Western Slide 10-6

Z Formula for the Difference in Two Sample Means

for n1 30, n2 30, and Independent Samples

Z Formula for the Difference in Two Sample Means

for n1 30, n2 30, and Independent Samples

ZX X

n n

1 2 1 2

1

2

1

2

2

2

© 2002 Thomson / South-Western Slide 10-7

Example: Hypothesis Testing for Differences Between Means (Part 1)

Example: Hypothesis Testing for Differences Between Means (Part 1)

Computer Analysts

24.10 25.00 24.25

23.75 22.70 21.75

24.25 21.30 22.00

22.00 22.55 18.00

23.50 23.25 23.50

22.80 22.10 22.70

24.00 24.25 21.50

23.85 23.50 23.80

24.20 22.75 25.60

22.90 23.80 24.10

23.20

23.55

Registered Nurses

20.75

23.80

22.00

21.85

24.16

21.10

23.30

24.00

21.75

21.50

20.40

23.25

22.75

23.00

21.25

20.00

21.75

20.5023.75

22.50

25.00

22.70

23.25

21.90

19.50

21.75

20.80

20.25

22.45

19.10

22.60

21.70

20.75

22.50

1

1

1

1

2

32

2314

1373

1885

nXSS

.

.

.

2

2

2

2

2

34

2199

1403

1968

nXSS

.

.

.

© 2002 Thomson / South-Western Slide 10-8

Example: Hypothesis Testing for Differences Between Means (Part 2)

Example: Hypothesis Testing for Differences Between Means (Part 2)

H

H

o

a

:

:

1 2

1 2

0

0

1 2X X

Rejection Region

Nonrejection Region

Critical Values

Rejection Region

1 2X X

2

01. 2

01.

© 2002 Thomson / South-Western Slide 10-9

Example: Hypothesis Testing for Differences Between Means (Part 3)

Example: Hypothesis Testing for Differences Between Means (Part 3)

Rejection Region

Nonrejection Region

Critical Values

Rejection Region

cZ 2 33.

2

01.

0 cZ 2 33.

2

01.

If Z < - 2.33 or Z > 2.33, reject H .

If - 2.33 Z 2.33, do not reject H .

o

o

© 2002 Thomson / South-Western Slide 10-10

Example: Hypothesis Testing for Differences between Means (Part 4)Example: Hypothesis Testing for

Differences between Means (Part 4)

ZX X

SnSn

1 2 1 2

1

2

1

2

2

2

2314 2199 0

188532

196834

336

. .

. ..

If Z < - 2.33 or Z > 2.33, reject H .

If - 2.33 Z 2.33, do not reject H .

o

o

Since Z = 3.36 > 2.33, reject H .o

Rejection Region

Nonrejection Region

Critical Values

Rejection Region

cZ 2 33.

2

01.

0 cZ 2 33.

2

01.

© 2002 Thomson / South-Western Slide 10-11

Demonstration Problem 10.1 (Part 1)Demonstration Problem 10.1 (Part 1)

H

H

o

a

:

:

1 2

1 2

0

0

Nonrejection Region

Critical Value

Rejection Region

.001

cZ 3 08. 0

© 2002 Thomson / South-Western Slide 10-12

Demonstration Problem 10.1 (Part 2)Demonstration Problem 10.1 (Part 2)

Nonrejection Region

Critical Value

Rejection Region

.001

cZ 3 08. 0

If Z < - 3.08, reject H .

If Z , do not reject H .

o

o 3 08.

1 2 1 2

2 2

1 2

1 2

2 2

3343 5568 09.40

87 761226 1716

ZX X

S Sn n

oSince Z = -9.40 < -3.08, reject H .

1

1

1

$3,343

$1,226

87

Women

XSn

2

2

2

$5,568

$1,716

76

Men

XSn

© 2002 Thomson / South-Western Slide 10-13

The t Test for Differencesin Population Means

The t Test for Differencesin Population Means

• Each of the two populations is normally distributed.

• The two samples are independent.• At least one of the samples is small, n < 30.

• The values of the population variances are unknown.

• The variances of the two populations are equal, 12 = 2

2

© 2002 Thomson / South-Western Slide 10-14

t Formula to Test the Difference in Means Assuming 1

2 = 22

t Formula to Test the Difference in Means Assuming 1

2 = 22

tX X

S n S nn n n n

1 2 1 2

1

2

1 2

2

2

1 2 1 2

1 1

21 1

© 2002 Thomson / South-Western Slide 10-15

Hernandez Manufacturing Company (Part 1)

Hernandez Manufacturing Company (Part 1)

H

H

o

a

:

:

1 2

1 2

0

0

If t < - 2.060 or t > 2.060, reject H .

If - 2.060 t 2.060, do not reject H .

o

o

2

05

2025

2 15 12 2 25

2 060

1 2

0 25 25

..

.. ,

df n nt

Rejection Region

Nonrejection Region

Critical Values

Rejection Region

2

025.

0 . , .025 25 2 060t

2

025.

. , .025 25 2 060t

© 2002 Thomson / South-Western Slide 10-16

Hernandez Manufacturing Company (Part 2)

Hernandez Manufacturing Company (Part 2)

Training Method A

56 51 45

47 52 43

42 53 52

50 42 48

47 44 44

Training Method B

59

52

53

54

57

56

55

64

53

65

53

57

1

1

1

2

15

47 73

19 495

nXS

.

.

2

2

2

2

12

56 5

18 273

nXS

.

.

© 2002 Thomson / South-Western Slide 10-17

Hernandez Manufacturing Company (Part 3)

Hernandez Manufacturing Company (Part 3)

Since t = -5.20 < -2.060, reject H .o

tX X

S n S nn n n n

1 2 1 2

1

2

1 2

2

2

1 2 1 2

1 1

21 1

47 73 5650 0

19 495 14 18 273 1115 12 2

115

112

520

. .

. .

.

If t < - 2.060 or t > 2.060, reject H .

If - 2.060 t 2.060, do not reject H .

o

o

© 2002 Thomson / South-Western Slide 10-18

Dependent SamplesDependent Samples

• Before and After Measurements on the same individual

• Studies of twins• Studies of

spouses

Individual

1

2

3

4

5

6

Before

32

11

21

17

30

38

After

39

15

35

13

41

39

d

-7

-4

-14

4

-11

-1

© 2002 Thomson / South-Western Slide 10-19

Formulas for Dependent SamplesFormulas for Dependent Samples

td D

ndf n

n

dS

1

number of pairs

d = sample difference in pairs

D = mean population difference

= standard deviation of sample difference

d = mean sample differencedS

dd

n

n

nn

dSd d

dd

2

2

2

1

1

© 2002 Thomson / South-Western Slide 10-20

Sampling Distribution of Differences in Sample Proportions

Sampling Distribution of Differences in Sample Proportions

For large samples

1.

2.

3. and

4. where q = 1 - p

the difference in sample proportions is normally distributed with

p and

p

1

1

2

2

nnnn

1

1

1

1

2

2

1 2

1 1

1

2 2

2

5

5

5

5

2

2

,

,

,

pqpq

P P

P Qn

P Qn

p

p

© 2002 Thomson / South-Western Slide 10-21

Z Formula for the Difference in Two Population ProportionsZ Formula for the Difference

in Two Population Proportions

Zp p P P

P Qn

P Qn

pp

1 2 1 2

1 1

1

2 2

2

1

2

proportion from sample 1

proportion from sample 2

size of sample 1

size of sample 2

proportion from population 1

proportion from population 2

1 -

1 -

1

2

1

2

1 1

2 2

nnPPQ PQ P

© 2002 Thomson / South-Western Slide 10-22

Z Formula to Test the Difference in Population Proportions

Z Formula to Test the Difference in Population Proportions

Z

P Q

P

Q P

p p P P

n nX Xn nn p n pn n

1 2 1 2

1 2

1 2

1 2

1 1 2 2

1 2

1 1

1

© 2002 Thomson / South-Western Slide 10-23

Testing the Difference in Population Proportions: Demonstration Problem 10.4

Testing the Difference in Population Proportions: Demonstration Problem 10.4

H

H

o

a

P PP P

:

:

1 2

1 2

0

0

If Z < - 2.575 or Z > 2.575, reject H .

If - 2.575 Z 2.575, do not reject H .

o

o

2

01

2005

2 575005

..

..Z

Rejection Region

Nonrejection Region

Critical Values

Rejection Region

2

005.

0cZ 2 575.

2

005.

cZ 2 575.

© 2002 Thomson / South-Western Slide 10-24

Demonstration Problem 10.4, continuedDemonstration Problem 10.4, continued

1

1

1

100

24

24

10024

nX

p

.

2

2

2

95

39

39

9541

nX

p

.

P X Xn n

1 2

1 2

24 39

100 95323.

Z

P Q

p p P P

n n

1 2 1 2

1 2

1 1

24 41 0

323 6771

100

1

95

17

0672 54

. .

. .

.

..

Since - 2.575 Z = - 2.54 2.575, do not reject H .o

© 2002 Thomson / South-Western Slide 10-25

Hypothesis Testing about the Difference in Two Population Variances

Hypothesis Testing about the Difference in Two Population Variances

1

1

22min

11

2

2

2

1

nn

SS

atordeno

numerator

df

df

F

F Test for Two Population Variances

© 2002 Thomson / South-Western Slide 10-26

Example: An F Distribution for 1 = 10 and 2 = 8

Example: An F Distribution for 1 = 10 and 2 = 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.00 1.00 2.00 3.00 4.00 5.00 6.00

© 2002 Thomson / South-Western Slide 10-27

A Portion of the F Distribution Table for = 0.025

A Portion of the F Distribution Table for = 0.025

Numerator Degrees of Freedom

DenominatorDegrees of Freedom

. , ,025 9 11F

1 2 3 4 5 6 7 8 91 647.79 799.48 864.15 899.60 921.83 937.11 948.20 956.64 963.282 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.393 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.474 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.905 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.686 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.527 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.828 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.369 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03

10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.7811 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.5912 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44

© 2002 Thomson / South-Western Slide 10-28

Hypothesis Test for Equality of Two Population Variances:

Sheet Metal Example (Part 1)

Hypothesis Test for Equality of Two Population Variances:

Sheet Metal Example (Part 1)

H

H

o

a

:

:

1

2

2

2

1

2

2

2

.025,9,11F 359.

If

If

F < 0.28 or F > 3.59 reject H .

0.28 F do reject H .

o

o

,

. , 359

= .975,11,9.025,9,11

F F1

1

3590 28

..

F

df

df

SS

nn

numerator

deno ator

1

2

2

2

1 1

2 2

1

1

min

0 05

10

12

1

2

.

nn

© 2002 Thomson / South-Western Slide 10-29

Sheet Metal Example (Part 2)Sheet Metal Example (Part 2)

Rejection Regions

Critical Values

. , , .025 9 11 359F

NonrejectionRegion

. , , .975 11 9 0 28F

If

If

F < 0.28 or F > 3.59 reject H .

0.28 F do reject H .

o

o

,

. , 359

© 2002 Thomson / South-Western Slide 10-30

Sheet Metal Example (Part 3)Sheet Metal Example (Part 3)

Machine 1

22.3 21.8 22.2

21.8 21.9 21.6

22.3 22.4

21.6 22.5

Machine 2

22.0

22.1

21.8

21.9

22.2

22.0

21.7

21.9

22.0

22.1

21.9

22.1

1

1

2

10

01138

nS

.

2

2

2

12

0 0202

nS

.FSS

1

2

2

2

01138

0 02025 63

.

..

.Hreject 3.59, = F > 5.63 =F Since oc