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Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 9
Two-Sample Tests

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Chap 9-1
Chapter Goals
After completing this chapter, you should be able to:


Test hypotheses for the difference between two
independent population means (standard deviations known
or unknown)



Test two means from related samples for the mean
difference



Complete a Z test for the difference between two
proportions



Use the F table to find critical F values

Complete an F test for
Statistics for Managers Using the difference between two
variances
Microsoft Excel, 4e © 2004
Chap 9-2
Prentice-Hall, Inc.

Two Sample Tests
Two Sample Tests

Population
Means,
Independent
Samples

Means,
Related
Samples

Population
Proportions

Population
Variances

Examples:
Group 1 vs.
independent
Group 2
Statistics for

Same group
before vs. after
treatment
Managers Using

Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Proportion 1 vs.
Proportion 2

Variance 1 vs.
Variance 2

Chap 9-3
Difference Between Two Means
Population means,
independent
samples

*

σ1 and σ2 known

Goal: Test hypotheses or form
a confidence interval for the
difference between two
population means, μ1 – μ2

σ1 and σ2 unknown

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

The point estimate for the
difference is

X1 – X2
Chap 9-4
Independent Samples
Population means,
independent
samples



*

Different data sources
 Unrelated
 Independent


σ1 and σ2 known
σ1 and σ2 unknown





Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Sample selected from one
population has no effect on
the sample selected from
the other population

Use the difference between 2
sample means
Use Z test or pooled variance
t test

Chap 9-5
Difference Between Two Means
Population means,
independent
samples

*

σ1 and σ2 known

Use a Z test statistic

σ1 and σ2 unknown

Use S to estimate unknown
σ , use a t test statistic and
pooled standard deviation

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 9-6
σ1 and σ2 Known
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown

Assumptions:

*

 Samples are randomly and
independently drawn
 population distributions are
normal or both sample sizes
are ≥ 30

 Population standard
Statistics for Managers Using
deviations are known
Microsoft Excel, 4e © 2004
Chap 9-7
Prentice-Hall, Inc.
σ1 and σ2 Known
(continued)

Population means,
independent
samples
σ1 and σ2 known

When σ1 and σ2 are known and
both populations are normal or
both sample sizes are at least 30,
the test statistic is a Z-value…

*

…and the standard error of
X1 – X2 is

σ1 and σ2 unknown

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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σ X1 − X2

2
1

2

σ
σ2
=
+
n1
n2

Chap 9-8
σ1 and σ2 Known
(continued)

Population means,
independent
samples
σ1 and σ2 known

The test statistic for
μ1 – μ2 is:

*

σ1 and σ2 unknown

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

(X
Z=

1

)

− X 2 − ( μ1 − μ2 )
2
1

2

σ
σ2
+
n1
n2

Chap 9-9
Hypothesis Tests for
Two Population Means
Two Population Means, Independent Samples
Lower-tail test:

Upper-tail test:

Two-tail test:

H0: μ1 ≥ μ2
H1: μ1 < μ2

H0: μ1 ≤ μ2
H1: μ1 > μ2

H0: μ1 = μ2
H1: μ1 ≠ μ2

i.e.,

i.e.,

i.e.,

H0: μ1 – μ2 ≥ 0
H1: μ1 – μ2 < 0

H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0

H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0

Statistics for Managers Using
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Chap 9-10
Hypothesis tests for μ1 – μ2
Two Population Means, Independent Samples
Lower-tail test:

Upper-tail test:

Two-tail test:

H0: μ1 – μ2 ≥ 0
H1: μ1 – μ2 < 0

H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0

H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0

α

α
-zα

Reject H0 if Z < -Zα

zα
Reject H0 if Z > Zα

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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α /2

α /2

-zα/2

zα/2

Reject H0 if Z < -Zα/2
or Z > Zα/2

Chap 9-11
Confidence Interval,
σ1 and σ2 Known
Population means,
independent
samples
σ1 and σ2 known

The confidence interval for
μ1 – μ2 is:

*

σ1 and σ2 unknown

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

(

)

2
1

2

σ
σ2
X1 − X 2 ± Z
+
n1
n2

Chap 9-12
σ1 and σ2 Unknown
Assumptions:

Population means,
independent
samples

 Samples are randomly and
independently drawn

σ1 and σ2 known
σ1 and σ2 unknown

*

Statistics for Managers Using
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 Populations are normally
distributed or both sample
sizes are at least 30
 Population variances are
unknown but assumed equal
Chap 9-13
σ1 and σ2 Unknown
(continued)

Forming interval
estimates:

Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown

*

 The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ

 the test statistic is a t value
Statistics for Managers Using with (n1 + n2 – 2) degrees
of freedom
Microsoft Excel, 4e © 2004
Chap 9-14
Prentice-Hall, Inc.
σ1 and σ2 Unknown
(continued)

Population means,
independent
samples

The pooled standard
deviation is

σ1 and σ2 known
σ1 and σ2 unknown

*

Statistics for Managers Using
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Sp =

( n1 − 1) S12 + ( n2 − 1) S2 2
(n1 − 1) + (n2 − 1)

Chap 9-15
σ1 and σ2 Unknown
(continued)

The test statistic for
μ1 – μ2 is:

Population means,
independent
samples

(X
t=

1

σ1 and σ2 known
σ1 and σ2 unknown

)

− X 2 − ( μ1 − μ2 )
1 1 
S  + 
n n 
2 
 1
2
p

*

Where t has (n1 + n2 – 2) d.f.,

Statistics for Managers Using
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and

S

2
p

( n1 − 1) S12 + ( n2 − 1) S2 2
=
(n1 − 1) + (n2 − 1)

Chap 9-16
Confidence Interval,
σ1 and σ2 Unknown
Population means,
independent
samples

The confidence interval for
μ1 – μ2 is:

σ1 and σ2 known
σ1 and σ2 unknown

(X

1

*

)

− X 2 ± t n1 +n2 -2

1 1 
S  + 
n n 
2 
 1
2
p

Where

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

S

2
p

( n1 − 1) S12 + ( n2 − 1) S2 2
=
(n1 − 1) + (n2 − 1)

Chap 9-17
Pooled Sp t Test: Example
You are a financial analyst for a brokerage firm. Is there
a difference in dividend yield between stocks listed on the
NYSE & NASDAQ? You collect the following data:

NYSE
Number
21
Sample mean
3.27
Sample std dev 1.30

NASDAQ
25
2.53
1.16

Assuming both populations are
approximately normal with
equal variances, is
there for Managers average
Statistics a difference in Using
yield (α = 0.05)?
Microsoft Excel, 4e © 2004
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Chap 9-18
Calculating the Test Statistic
The test statistic is:

(X
t=

1

)

− X 2 − ( μ1 − μ2 )
1 1
S  + 
n n 
2 
 1

=

2
p

( 3.27 − 2.53 ) − 0
1 
 1
1.5021 +

 21 25 

( n1 − 1)S12 + ( n2 − 1)S2 2 = ( 21 − 1)1.30 2 + ( 25 − 1)1.16 2
S2 =
p

(n1 − 1) + (n2 − 1)

Statistics for Managers Using
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(21 - 1) + (25 − 1)

= 2.040

= 1.5021

Chap 9-19
Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
α = 0.05
df = 21 + 25 - 2 = 44
Critical Values: t = ± 2.0154

Test Statistic:
3.27 − 2.53
t=
= 2.040
1 
 1
1.5021  +

 21 25 
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Reject H0

.025

-2.0154

Reject H0

.025

0 2.0154

t

2.040

Decision:
Reject H0 at α = 0.05
Conclusion:
There is evidence of a
difference in means.
Chap 9-20
Related Samples
Tests Means of 2 Related Populations
Related
samples





Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:

D = X1 - X2
Eliminates Variation Among Subjects
 Assumptions:
 Both Populations Are Normally Distributed
 Or, if Not Normal, use large samples
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 9-21
Prentice-Hall, Inc.

Mean Difference, σD Known
The ith paired difference is Di , where
Related
samples

Di = X1i - X2i
The point estimate for
the population mean
paired difference is D :

n

D=

∑D
i=
1

i

n

Suppose the population standard
deviation of the difference
scores, σD, is known

Statistics for Managers Using
n is the number of pairs in the paired sample
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Chap 9-22
Mean Difference, σD Known
(continued)

Paired
samples

The test statistic for the mean
difference is a Z value:

D − μD
Z=
σD
n
Where
μD = hypothesized mean difference
σD = population standard dev. of differences
Managers Using
n = the sample size (number of pairs)

Statistics for
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-23
Confidence Interval, σD Known
Paired
samples

The confidence interval for D is

σD
D±Z
n
Where
n = the sample size
(number of pairs in the paired sample)

Statistics for Managers Using
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Chap 9-24
Mean Difference, σD Unknown
Related
samples

If σD is unknown, we can estimate the
unknown population standard deviation
with a sample standard deviation:
The sample standard
deviation is

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

n

SD =

(Di − D)2
∑
i=1

n −1

Chap 9-25
Mean Difference, σD Unknown
(continued)

Paired
samples

The test statistic for D is now a
t statistic, with n-1 d.f.:

D − μD
t=
SD
n
n

Where t has n - 1 d.f.
and
Statistics for Managers Using SD is:
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SD =

∑ (D
i=1

i

− D)

2

n−1

Chap 9-26
Confidence Interval, σD Unknown
Paired
samples

The confidence interval for D is

SD
D ± t n−1
n
n

where

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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SD =

∑ (D − D)
i=1

2

i

n −1

Chap 9-27
Hypothesis Testing for
Mean Difference, σD Unknown
Paired Samples
Lower-tail test:

Upper-tail test:

Two-tail test:

H0: μD ≥ 0
H1: μD < 0

H0: μD ≤ 0
H1: μD > 0

H0: μD = 0
H1: μD ≠ 0

α

α
-tα

Reject H if t < -t

tα
Reject H if t > t

α
α
Statistics 0 Managers Using 0
for
Microsoft Excel, 4e © Where t has n - 1 d.f.
2004
Prentice-Hall, Inc.

α /2

α /2

-tα/2

tα/2

Reject H0 if t < -tα/2
or t > tα/2

Chap 9-28
Paired Samples Example
Assume you send your salespeople to a “customer
service” training workshop. Is the training effective?
You collect the following data:


Number of Complaints:
(2) - (1)
Salesperson Before (1) After (2)
Difference, Di
C.B.
T.F.
M.H.
R.K.
M.O.

6
20
3
0
4

4
6
2
0
0

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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- 2
-14
- 1
0
- 4
-21

D =

Σ Di
n

= -4.2
SD =

∑ (D − D)
i

n −1

= 5.67

Chap 9-29

2
Paired Samples: Solution
 Has the training made a difference in the number of

complaints (at the 0.01 level)?
H0: μD = 0
H1: μD ≠ 0
α = .01

D = - 4.2

Critical Value = ± 4.604
d.f. = n - 1 = 4

Test Statistic:

D − for Managers
Statistics μD = − 4.2 − 0 Using
t=
= − 1.66
SD / n 5.67/ 2004
Microsoft Excel, 4e © 5
Prentice-Hall, Inc.

Reject

Reject

α/2

α/2

- 4.604

4.604

- 1.66

Decision: Do not reject H0
(t stat is not in the reject region)

Conclusion: There is not a
significant change in the
number of complaints.
Chap 9-30
Two Population Proportions
Population
proportions

Goal: test a hypothesis or form a
confidence interval for the difference
between two population proportions,
p1 – p2
Assumptions:
n1p1 ≥ 5 , n1(1-p1) ≥ 5
n2p2 ≥ 5 , n2(1-p2) ≥ 5

The point estimate for
Statistics for Managers Using is
the difference
Microsoft Excel, 4e © 2004
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ps1 − ps2
Chap 9-31
Two Population Proportions
Population
proportions

Since we begin by assuming the null
hypothesis is true, we assume p1 = p2
and pool the two ps estimates
The pooled estimate for the
overall proportion is:

X1 + X 2
p=
n1 + n2
where X1 and X2 are the numbers from
Statistics for Managers Using 1 and 2 with the characteristic of
samples
Microsoft Excel, 4e © 2004
interest

Prentice-Hall, Inc.

Chap 9-32
Two Population Proportions
(continued)

Population
proportions

The test statistic for
p1 – p2 is a Z statistic:

(p
Z=

s1

)

− p s2 − ( p1 − p 2 )

1 1
p (1 − p)  + 
n n 
2 
 1

X
Statistics for ManagerspUsing+ X 2 , p s = X1 , p s = X 2
where
= 1
n1
n2
Microsoft Excel, 4e © 2004n1 + n2
Chap 9-33
Prentice-Hall, Inc.
1

2
Confidence Interval for
Two Population Proportions
Population
proportions

(p

s1

The confidence interval for
p1 – p2 is:

)

− p s2 ± Z

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

p s1 (1 − p s1 )
n1

+

p s2 (1 − p s2 )
n2
Chap 9-34
Hypothesis Tests for
Two Population Proportions
Population proportions
Lower-tail test:

Upper-tail test:

Two-tail test:

H0: p1 ≥ p2
H1: p1 < p2

H0: p1 ≤ p2
H1: p1 > p2

H0: p1 = p2
H1: p1 ≠ p2

i.e.,

i.e.,

i.e.,

H0: p1 – p2 ≥ 0
H1: p1 – p2 < 0

H0: p1 – p2 ≤ 0
H1: p1 – p2 > 0

H0: p1 – p2 = 0
H1: p1 – p2 ≠ 0

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 9-35
Hypothesis Tests for
Two Population Proportions

(continued)

Population proportions
Lower-tail test:

Upper-tail test:

Two-tail test:

H0: p1 – p2 ≥ 0
H1: p1 – p2 < 0

H0: p1 – p2 ≤ 0
H1: p1 – p2 > 0

H0: p1 – p2 = 0
H1: p1 – p2 ≠ 0

α

α
-zα

Reject H0 if Z < -Zα

zα
Reject H0 if Z > Zα

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

α /2

α /2

-zα/2

zα/2

Reject H0 if Z < -Zα/2
or Z > Zα/2

Chap 9-36
Example:
Two population Proportions
Is there a significant difference between the
proportion of men and the proportion of
women who will vote Yes on Proposition A?



In a random sample, 36 of 72 men and 31 of
50 women indicated they would vote Yes



Test at the .05 level of significance

Statistics for Managers Using
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Chap 9-37
Example:
Two population Proportions

(continued)

The hypothesis test is:



H0: p1 – p2 = 0 (the two proportions are equal)
H1: p1 – p2 ≠ 0 (there is a significant difference between proportions)


The sample proportions are:


Men:



Women:

ps1 = 36/72 = .50
ps2 = 31/50 = .62

 The pooled estimate for the overall proportion is:

X1 + X 2 36 + 31 67
p=
=
=
= .549
Statistics for Managers Using + 50 122
n1 + n2
72
Microsoft Excel, 4e © 2004
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Chap 9-38
Example:
Two population Proportions

(continued)

Reject H0

The test statistic for p1 – p2 is:

(p
z=
=

s1

)

− p s2 − ( p1 − p 2 )

1
1
p (1 − p)  + 
n n 
2 
 1

( .50 − .62) − ( 0)
1 
 1
.549 (1 − .549) 
+

72 50 


Reject H0

.025

.025

-1.96
-1.31

= − 1.31

Critical Values = ±1.96
StatisticsFor Managers Using
for α = .05
Microsoft Excel, 4e © 2004
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1.96

Decision: Do not reject H0
Conclusion: There is not
significant evidence of a
difference in proportions
who will vote yes between
men and women. 9-39
Chap
Hypothesis Tests for Variances
Tests for Two
Population
Variances
F test statistic

*

H0: σ12 = σ22
H1: σ12 ≠ σ22

Two-tail test

H0: σ12 ≥ σ22
H1: σ12 < σ22

Lower-tail test

H0: σ12 ≤ σ22
H1: σ12 > σ22
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Upper-tail test

Chap 9-40
Hypothesis Tests for Variances
(continued)

Tests for Two
Population
Variances
F test statistic

The F test statistic is:

*

2
1
2
2

S
F=
S

2
S1 = Variance of Sample 1

n1 - 1 = numerator degrees of freedom

S 2 = Variance of Sample 2
2
Statistics for Managers Using n2 - 1 = denominator degrees of freedom
Microsoft Excel, 4e © 2004
Chap 9-41
Prentice-Hall, Inc.
The F Distribution


The F critical value is found from the F table



The are two appropriate degrees of freedom:
numerator and denominator
2
S1
F= 2
S2



where

df1 = n1 – 1 ; df2 = n2 – 1

In the F table,


numerator degrees of freedom determine the column



denominator degrees of freedom determine the row

Statistics for Managers Using
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Chap 9-42
Finding the Rejection Region
H0: σ12 ≥ σ22
H1: σ12 < σ22

α

α/2

0
Reject
H0

FL

F

Do not
reject H0

Reject
H0

H0: σ12 ≤ σ22
H1: σ12 > σ22
α

0
Do
StatisticsnotH Managers Using F
for F Reject H
reject
U
Microsoft Excel, 4e © 2004
Reject H if F > F
Prentice-Hall, Inc.
0

0

U

α/2

0

Reject H0 if F < FL

0

H0: σ12 = σ22
H1: σ12 ≠ σ22

FL

Do not
reject H0

FU

rejection
region for a
two-tail test is:


Reject H0

2
S1
F = 2 > FU
S2
2
S1
F = 2 < FL
S2

Chap 9-43

F
Finding the Rejection Region
(continued)
α/2

H0: σ12 = σ22
H1: σ12 ≠ σ22
α/2

0

Reject
H0

To find the critical F values:
1. Find FU from the F table
for n1 – 1 numerator and
n2 – 1 denominator
degrees of Managers
Statistics forfreedom

Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

FL

Do not
reject H0

FU

Reject H0

F

1
2. Find FL using the formula: FL =
FU*
Where FU* is from the F table with
n2 – 1 numerator and n1 – 1
denominator degrees of freedom
(i.e., switch the d.f. from FU)

Chap 9-44
F Test: An Example
You are a financial analyst for a brokerage firm. You
want to compare dividend yields between stocks listed on
the NYSE & NASDAQ. You collect the following data :
NYSE
NASDAQ
Number
21
25
Mean
3.27
2.53
Std dev
1.30
1.16
Is there a difference in the
variances between the NYSE
Statistics for ManagersαUsing level?
& NASDAQ at the = 0.05
Microsoft Excel, 4e © 2004
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Chap 9-45
F Test: Example Solution


Form the hypothesis test:
H0: σ21 – σ22 = 0 (there is no difference between variances)
H1: σ21 – σ22 ≠ 0



(there is a difference between variances)

Find the F critical values for α = .05:

FU:
 Numerator:




n1 – 1 = 21 – 1 = 20 d.f.

Denominator:


FL:
 Numerator:

n2 – 1 = 25 – 1 = 24 d.f.

Statistics UforF.025, 20, 24 = 2.33
F = Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.





n2 – 1 = 25 – 1 = 24 d.f.

Denominator:


n1 – 1 = 21 – 1 = 20 d.f.

FL = 1/F.025, 24, 20 = 1/2.41
= .41
Chap 9-46
F Test: Example Solution
(continued)


The test statistic is:
2
1
2
2

H0: σ12 = σ22
H1: σ12 ≠ σ22

2

S
1.30
F=
=
= 1.256
2
S
1.16

α/2 = .025
0

α/2 = .025

Reject H0


F = 1.256 is not in the rejection
region, so we do not reject H0

Do not
reject H0

FL=0.41

Conclusion: There is not sufficient evidence
of a difference in variances
Statistics for Managers Using at α = .05
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Reject H0

FU=2.33



Chap 9-47

F
Two-Sample Tests in EXCEL
For independent samples:


Independent sample Z test with variances known:


Tools | data analysis | z-test: two sample for means

For paired samples (t test):


Tools | data analysis… | t-test: paired two sample for means

For variances…


F test for two variances:


Tools | data analysis | F-test: two sample for variances

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-48
Two-Sample Tests in PHStat

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-49
Sample PHStat Output

Input

Output

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-50
Sample PHStat Output
(continued)

Input

Output

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-51
Chapter Summary


Compared two independent samples







Performed Z test for the differences in two means
Performed pooled variance t test for the differences
in two means
Formed confidence intervals for the differences
between two means

Compared two related samples (paired
samples)

Performed paired sample Z and t tests for the mean
difference
 Formed confidence intervals for the paired difference
Statistics for Managers Using


Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-52
Chapter Summary
(continued)


Compared two population proportions




Formed confidence intervals for the difference
between two population proportions
Performed Z-test for two population proportions



Performed F tests for the difference between
two population variances



Used the F table to find F critical values

Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

Chap 9-53

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Chap09 2 sample test

  • 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 9 Two-Sample Tests Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-1
  • 2. Chapter Goals After completing this chapter, you should be able to:  Test hypotheses for the difference between two independent population means (standard deviations known or unknown)  Test two means from related samples for the mean difference  Complete a Z test for the difference between two proportions  Use the F table to find critical F values Complete an F test for Statistics for Managers Using the difference between two variances Microsoft Excel, 4e © 2004 Chap 9-2 Prentice-Hall, Inc. 
  • 3. Two Sample Tests Two Sample Tests Population Means, Independent Samples Means, Related Samples Population Proportions Population Variances Examples: Group 1 vs. independent Group 2 Statistics for Same group before vs. after treatment Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 Chap 9-3
  • 4. Difference Between Two Means Population means, independent samples * σ1 and σ2 known Goal: Test hypotheses or form a confidence interval for the difference between two population means, μ1 – μ2 σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. The point estimate for the difference is X1 – X2 Chap 9-4
  • 5. Independent Samples Population means, independent samples  * Different data sources  Unrelated  Independent  σ1 and σ2 known σ1 and σ2 unknown   Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Sample selected from one population has no effect on the sample selected from the other population Use the difference between 2 sample means Use Z test or pooled variance t test Chap 9-5
  • 6. Difference Between Two Means Population means, independent samples * σ1 and σ2 known Use a Z test statistic σ1 and σ2 unknown Use S to estimate unknown σ , use a t test statistic and pooled standard deviation Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-6
  • 7. σ1 and σ2 Known Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown Assumptions: *  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are ≥ 30  Population standard Statistics for Managers Using deviations are known Microsoft Excel, 4e © 2004 Chap 9-7 Prentice-Hall, Inc.
  • 8. σ1 and σ2 Known (continued) Population means, independent samples σ1 and σ2 known When σ1 and σ2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a Z-value… * …and the standard error of X1 – X2 is σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. σ X1 − X2 2 1 2 σ σ2 = + n1 n2 Chap 9-8
  • 9. σ1 and σ2 Known (continued) Population means, independent samples σ1 and σ2 known The test statistic for μ1 – μ2 is: * σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. (X Z= 1 ) − X 2 − ( μ1 − μ2 ) 2 1 2 σ σ2 + n1 n2 Chap 9-9
  • 10. Hypothesis Tests for Two Population Means Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 ≥ μ2 H1: μ1 < μ2 H0: μ1 ≤ μ2 H1: μ1 > μ2 H0: μ1 = μ2 H1: μ1 ≠ μ2 i.e., i.e., i.e., H0: μ1 – μ2 ≥ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-10
  • 11. Hypothesis tests for μ1 – μ2 Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 – μ2 ≥ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 α α -zα Reject H0 if Z < -Zα zα Reject H0 if Z > Zα Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α /2 α /2 -zα/2 zα/2 Reject H0 if Z < -Zα/2 or Z > Zα/2 Chap 9-11
  • 12. Confidence Interval, σ1 and σ2 Known Population means, independent samples σ1 and σ2 known The confidence interval for μ1 – μ2 is: * σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ( ) 2 1 2 σ σ2 X1 − X 2 ± Z + n1 n2 Chap 9-12
  • 13. σ1 and σ2 Unknown Assumptions: Population means, independent samples  Samples are randomly and independently drawn σ1 and σ2 known σ1 and σ2 unknown * Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown but assumed equal Chap 9-13
  • 14. σ1 and σ2 Unknown (continued) Forming interval estimates: Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown *  The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ  the test statistic is a t value Statistics for Managers Using with (n1 + n2 – 2) degrees of freedom Microsoft Excel, 4e © 2004 Chap 9-14 Prentice-Hall, Inc.
  • 15. σ1 and σ2 Unknown (continued) Population means, independent samples The pooled standard deviation is σ1 and σ2 known σ1 and σ2 unknown * Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Sp = ( n1 − 1) S12 + ( n2 − 1) S2 2 (n1 − 1) + (n2 − 1) Chap 9-15
  • 16. σ1 and σ2 Unknown (continued) The test statistic for μ1 – μ2 is: Population means, independent samples (X t= 1 σ1 and σ2 known σ1 and σ2 unknown ) − X 2 − ( μ1 − μ2 ) 1 1  S  +  n n  2   1 2 p * Where t has (n1 + n2 – 2) d.f., Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. and S 2 p ( n1 − 1) S12 + ( n2 − 1) S2 2 = (n1 − 1) + (n2 − 1) Chap 9-16
  • 17. Confidence Interval, σ1 and σ2 Unknown Population means, independent samples The confidence interval for μ1 – μ2 is: σ1 and σ2 known σ1 and σ2 unknown (X 1 * ) − X 2 ± t n1 +n2 -2 1 1  S  +  n n  2   1 2 p Where Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. S 2 p ( n1 − 1) S12 + ( n2 − 1) S2 2 = (n1 − 1) + (n2 − 1) Chap 9-17
  • 18. Pooled Sp t Test: Example You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE Number 21 Sample mean 3.27 Sample std dev 1.30 NASDAQ 25 2.53 1.16 Assuming both populations are approximately normal with equal variances, is there for Managers average Statistics a difference in Using yield (α = 0.05)? Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-18
  • 19. Calculating the Test Statistic The test statistic is: (X t= 1 ) − X 2 − ( μ1 − μ2 ) 1 1 S  +  n n  2   1 = 2 p ( 3.27 − 2.53 ) − 0 1   1 1.5021 +   21 25  ( n1 − 1)S12 + ( n2 − 1)S2 2 = ( 21 − 1)1.30 2 + ( 25 − 1)1.16 2 S2 = p (n1 − 1) + (n2 − 1) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. (21 - 1) + (25 − 1) = 2.040 = 1.5021 Chap 9-19
  • 20. Solution H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) α = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Test Statistic: 3.27 − 2.53 t= = 2.040 1   1 1.5021  +   21 25  Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Reject H0 .025 -2.0154 Reject H0 .025 0 2.0154 t 2.040 Decision: Reject H0 at α = 0.05 Conclusion: There is evidence of a difference in means. Chap 9-20
  • 21. Related Samples Tests Means of 2 Related Populations Related samples    Paired or matched samples Repeated measures (before/after) Use difference between paired values: D = X1 - X2 Eliminates Variation Among Subjects  Assumptions:  Both Populations Are Normally Distributed  Or, if Not Normal, use large samples Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 9-21 Prentice-Hall, Inc. 
  • 22. Mean Difference, σD Known The ith paired difference is Di , where Related samples Di = X1i - X2i The point estimate for the population mean paired difference is D : n D= ∑D i= 1 i n Suppose the population standard deviation of the difference scores, σD, is known Statistics for Managers Using n is the number of pairs in the paired sample Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-22
  • 23. Mean Difference, σD Known (continued) Paired samples The test statistic for the mean difference is a Z value: D − μD Z= σD n Where μD = hypothesized mean difference σD = population standard dev. of differences Managers Using n = the sample size (number of pairs) Statistics for Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-23
  • 24. Confidence Interval, σD Known Paired samples The confidence interval for D is σD D±Z n Where n = the sample size (number of pairs in the paired sample) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-24
  • 25. Mean Difference, σD Unknown Related samples If σD is unknown, we can estimate the unknown population standard deviation with a sample standard deviation: The sample standard deviation is Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. n SD = (Di − D)2 ∑ i=1 n −1 Chap 9-25
  • 26. Mean Difference, σD Unknown (continued) Paired samples The test statistic for D is now a t statistic, with n-1 d.f.: D − μD t= SD n n Where t has n - 1 d.f. and Statistics for Managers Using SD is: Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. SD = ∑ (D i=1 i − D) 2 n−1 Chap 9-26
  • 27. Confidence Interval, σD Unknown Paired samples The confidence interval for D is SD D ± t n−1 n n where Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. SD = ∑ (D − D) i=1 2 i n −1 Chap 9-27
  • 28. Hypothesis Testing for Mean Difference, σD Unknown Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μD ≥ 0 H1: μD < 0 H0: μD ≤ 0 H1: μD > 0 H0: μD = 0 H1: μD ≠ 0 α α -tα Reject H if t < -t tα Reject H if t > t α α Statistics 0 Managers Using 0 for Microsoft Excel, 4e © Where t has n - 1 d.f. 2004 Prentice-Hall, Inc. α /2 α /2 -tα/2 tα/2 Reject H0 if t < -tα/2 or t > tα/2 Chap 9-28
  • 29. Paired Samples Example Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data:  Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, Di C.B. T.F. M.H. R.K. M.O. 6 20 3 0 4 4 6 2 0 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. - 2 -14 - 1 0 - 4 -21 D = Σ Di n = -4.2 SD = ∑ (D − D) i n −1 = 5.67 Chap 9-29 2
  • 30. Paired Samples: Solution  Has the training made a difference in the number of complaints (at the 0.01 level)? H0: μD = 0 H1: μD ≠ 0 α = .01 D = - 4.2 Critical Value = ± 4.604 d.f. = n - 1 = 4 Test Statistic: D − for Managers Statistics μD = − 4.2 − 0 Using t= = − 1.66 SD / n 5.67/ 2004 Microsoft Excel, 4e © 5 Prentice-Hall, Inc. Reject Reject α/2 α/2 - 4.604 4.604 - 1.66 Decision: Do not reject H0 (t stat is not in the reject region) Conclusion: There is not a significant change in the number of complaints. Chap 9-30
  • 31. Two Population Proportions Population proportions Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, p1 – p2 Assumptions: n1p1 ≥ 5 , n1(1-p1) ≥ 5 n2p2 ≥ 5 , n2(1-p2) ≥ 5 The point estimate for Statistics for Managers Using is the difference Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ps1 − ps2 Chap 9-31
  • 32. Two Population Proportions Population proportions Since we begin by assuming the null hypothesis is true, we assume p1 = p2 and pool the two ps estimates The pooled estimate for the overall proportion is: X1 + X 2 p= n1 + n2 where X1 and X2 are the numbers from Statistics for Managers Using 1 and 2 with the characteristic of samples Microsoft Excel, 4e © 2004 interest Prentice-Hall, Inc. Chap 9-32
  • 33. Two Population Proportions (continued) Population proportions The test statistic for p1 – p2 is a Z statistic: (p Z= s1 ) − p s2 − ( p1 − p 2 ) 1 1 p (1 − p)  +  n n  2   1 X Statistics for ManagerspUsing+ X 2 , p s = X1 , p s = X 2 where = 1 n1 n2 Microsoft Excel, 4e © 2004n1 + n2 Chap 9-33 Prentice-Hall, Inc. 1 2
  • 34. Confidence Interval for Two Population Proportions Population proportions (p s1 The confidence interval for p1 – p2 is: ) − p s2 ± Z Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. p s1 (1 − p s1 ) n1 + p s2 (1 − p s2 ) n2 Chap 9-34
  • 35. Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: p1 ≥ p2 H1: p1 < p2 H0: p1 ≤ p2 H1: p1 > p2 H0: p1 = p2 H1: p1 ≠ p2 i.e., i.e., i.e., H0: p1 – p2 ≥ 0 H1: p1 – p2 < 0 H0: p1 – p2 ≤ 0 H1: p1 – p2 > 0 H0: p1 – p2 = 0 H1: p1 – p2 ≠ 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-35
  • 36. Hypothesis Tests for Two Population Proportions (continued) Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: p1 – p2 ≥ 0 H1: p1 – p2 < 0 H0: p1 – p2 ≤ 0 H1: p1 – p2 > 0 H0: p1 – p2 = 0 H1: p1 – p2 ≠ 0 α α -zα Reject H0 if Z < -Zα zα Reject H0 if Z > Zα Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α /2 α /2 -zα/2 zα/2 Reject H0 if Z < -Zα/2 or Z > Zα/2 Chap 9-36
  • 37. Example: Two population Proportions Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?  In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes  Test at the .05 level of significance Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-37
  • 38. Example: Two population Proportions (continued) The hypothesis test is:  H0: p1 – p2 = 0 (the two proportions are equal) H1: p1 – p2 ≠ 0 (there is a significant difference between proportions)  The sample proportions are:  Men:  Women: ps1 = 36/72 = .50 ps2 = 31/50 = .62  The pooled estimate for the overall proportion is: X1 + X 2 36 + 31 67 p= = = = .549 Statistics for Managers Using + 50 122 n1 + n2 72 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-38
  • 39. Example: Two population Proportions (continued) Reject H0 The test statistic for p1 – p2 is: (p z= = s1 ) − p s2 − ( p1 − p 2 ) 1 1 p (1 − p)  +  n n  2   1 ( .50 − .62) − ( 0) 1   1 .549 (1 − .549)  +  72 50   Reject H0 .025 .025 -1.96 -1.31 = − 1.31 Critical Values = ±1.96 StatisticsFor Managers Using for α = .05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1.96 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. 9-39 Chap
  • 40. Hypothesis Tests for Variances Tests for Two Population Variances F test statistic * H0: σ12 = σ22 H1: σ12 ≠ σ22 Two-tail test H0: σ12 ≥ σ22 H1: σ12 < σ22 Lower-tail test H0: σ12 ≤ σ22 H1: σ12 > σ22 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Upper-tail test Chap 9-40
  • 41. Hypothesis Tests for Variances (continued) Tests for Two Population Variances F test statistic The F test statistic is: * 2 1 2 2 S F= S 2 S1 = Variance of Sample 1 n1 - 1 = numerator degrees of freedom S 2 = Variance of Sample 2 2 Statistics for Managers Using n2 - 1 = denominator degrees of freedom Microsoft Excel, 4e © 2004 Chap 9-41 Prentice-Hall, Inc.
  • 42. The F Distribution  The F critical value is found from the F table  The are two appropriate degrees of freedom: numerator and denominator 2 S1 F= 2 S2  where df1 = n1 – 1 ; df2 = n2 – 1 In the F table,  numerator degrees of freedom determine the column  denominator degrees of freedom determine the row Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-42
  • 43. Finding the Rejection Region H0: σ12 ≥ σ22 H1: σ12 < σ22 α α/2 0 Reject H0 FL F Do not reject H0 Reject H0 H0: σ12 ≤ σ22 H1: σ12 > σ22 α 0 Do StatisticsnotH Managers Using F for F Reject H reject U Microsoft Excel, 4e © 2004 Reject H if F > F Prentice-Hall, Inc. 0 0 U α/2 0 Reject H0 if F < FL 0 H0: σ12 = σ22 H1: σ12 ≠ σ22 FL Do not reject H0 FU rejection region for a two-tail test is:  Reject H0 2 S1 F = 2 > FU S2 2 S1 F = 2 < FL S2 Chap 9-43 F
  • 44. Finding the Rejection Region (continued) α/2 H0: σ12 = σ22 H1: σ12 ≠ σ22 α/2 0 Reject H0 To find the critical F values: 1. Find FU from the F table for n1 – 1 numerator and n2 – 1 denominator degrees of Managers Statistics forfreedom Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. FL Do not reject H0 FU Reject H0 F 1 2. Find FL using the formula: FL = FU* Where FU* is from the F table with n2 – 1 numerator and n1 – 1 denominator degrees of freedom (i.e., switch the d.f. from FU) Chap 9-44
  • 45. F Test: An Example You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data : NYSE NASDAQ Number 21 25 Mean 3.27 2.53 Std dev 1.30 1.16 Is there a difference in the variances between the NYSE Statistics for ManagersαUsing level? & NASDAQ at the = 0.05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-45
  • 46. F Test: Example Solution  Form the hypothesis test: H0: σ21 – σ22 = 0 (there is no difference between variances) H1: σ21 – σ22 ≠ 0  (there is a difference between variances) Find the F critical values for α = .05: FU:  Numerator:   n1 – 1 = 21 – 1 = 20 d.f. Denominator:  FL:  Numerator: n2 – 1 = 25 – 1 = 24 d.f. Statistics UforF.025, 20, 24 = 2.33 F = Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.   n2 – 1 = 25 – 1 = 24 d.f. Denominator:  n1 – 1 = 21 – 1 = 20 d.f. FL = 1/F.025, 24, 20 = 1/2.41 = .41 Chap 9-46
  • 47. F Test: Example Solution (continued)  The test statistic is: 2 1 2 2 H0: σ12 = σ22 H1: σ12 ≠ σ22 2 S 1.30 F= = = 1.256 2 S 1.16 α/2 = .025 0 α/2 = .025 Reject H0  F = 1.256 is not in the rejection region, so we do not reject H0 Do not reject H0 FL=0.41 Conclusion: There is not sufficient evidence of a difference in variances Statistics for Managers Using at α = .05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Reject H0 FU=2.33  Chap 9-47 F
  • 48. Two-Sample Tests in EXCEL For independent samples:  Independent sample Z test with variances known:  Tools | data analysis | z-test: two sample for means For paired samples (t test):  Tools | data analysis… | t-test: paired two sample for means For variances…  F test for two variances:  Tools | data analysis | F-test: two sample for variances Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-48
  • 49. Two-Sample Tests in PHStat Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-49
  • 50. Sample PHStat Output Input Output Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-50
  • 51. Sample PHStat Output (continued) Input Output Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-51
  • 52. Chapter Summary  Compared two independent samples     Performed Z test for the differences in two means Performed pooled variance t test for the differences in two means Formed confidence intervals for the differences between two means Compared two related samples (paired samples) Performed paired sample Z and t tests for the mean difference  Formed confidence intervals for the paired difference Statistics for Managers Using  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-52
  • 53. Chapter Summary (continued)  Compared two population proportions   Formed confidence intervals for the difference between two population proportions Performed Z-test for two population proportions  Performed F tests for the difference between two population variances  Used the F table to find F critical values Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-53