This chapter discusses two-sample tests, including tests for the difference between two independent population means, the difference between two related (paired) sample means, the difference between two population proportions, and the difference between two variances. It provides the formulas and procedures for conducting Z tests, t tests, and F tests for these comparisons in situations where the population standard deviations are both known and unknown. The goal is to test hypotheses about differences between parameters of two populations or to construct confidence intervals for these differences.

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
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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
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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
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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
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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
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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
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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
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(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
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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
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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

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
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(
)
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
*
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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

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
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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.,
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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
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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)?
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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)
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(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
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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

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
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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
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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
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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)
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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
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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
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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
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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
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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

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
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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
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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
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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
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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
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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
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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
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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
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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
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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

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
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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
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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
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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
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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
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

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