for Statistics Managers
Using Microsoft Excel
5th Edition
Chapter 10
Two-Sample Tests
Statistics for Managers Using Microsoft Excel, 5e 2008 Prentice-Hall, Inc.
Chap 10-1
Learning Objectives
In this chapter, you learn how to use hypothesis
testing for comparing the difference between:
The means of two independent populations
The means of two related populations
The proportions of two independent
populations
The variances of two independent
populations
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-2
Two-Sample Tests Overview
Two Sample Tests
Independent
Population
Means
Means,
Related
Populations
Independent
Population
Proportions
Same group
before vs. after
treatment
Proportion 1vs.
Proportion 2
Independent
Population
Variances
Examples
Group 1 vs.
Group 2
Statistics for Managers Using Microsoft Excel, 5e
Variance 1 vs.
Variance 2
Chap 10-3
Two-Sample Tests
Independent
Population Means
1 and 2 known
1 and 2 unknown
Goal: Test hypothesis or form
a confidence interval for the
difference between two
population means, 1 2
The point estimate for the
difference between sample
means:
X1 X2
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-4
Two-Sample Tests
Independent Populations
Different data sources
Independent
Population Means
1 and 2 known
1 and 2 unknown
Independent: 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, pooled variance t
test, or separate-variance t test
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-5
Two-Sample Tests
Independent Populations
Independent
Population Means
1 and 2 known
1 and 2 unknown
Use a Z test statistic
Use S to estimate unknown ,
use a t test statistic
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-6
Two-Sample Tests
Independent Populations
Independent
Population Means
1 and 2 known
Assumptions:
Samples are randomly and
independently drawn
population distributions are
normal
1 and 2 unknown
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-7
Two-Sample Tests
Independent Populations
Independent
Population Means
When 1 and 2 are known and both
populations are normal, the test
statistic is a Z-value and the
standard error of X1 X2 is
1 and 2 known
2
1 and 2 unknown
X1 X 2
Statistics for Managers Using Microsoft Excel, 5e
2
1 2
=
+
n1 n 2
Chap 10-8
Two-Sample Tests
Independent Populations
Independent
Population Means
1 and 2 known
The test statistic is:
( X X )(
Z=
1
1 and 2 unknown
Statistics for Managers Using Microsoft Excel, 5e
2
2
1
2 )
2
1 2
+
n1 n 2
Chap 10-9
Two-Sample Tests
Independent Populations
Two Independent Populations, Comparing Means
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: 1 2
H1: 1 < 2
H0: 1 2
H1: 1 > 2
H0: 1 = 2
H 1 : 1 2
i.e.,
i.e.,
i.e.,
H0: 1 2 0
H1: 1 2 < 0
H0: 1 2 0
H 1 : 1 2 > 0
H0: 1 2 = 0
H1: 1 2 0
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-10
Two-Sample Tests
Independent Populations
Two Independent Populations, Comparing Means
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: 1 2 0
H1: 1 2 < 0
H0: 1 2 0
H 1 : 1 2 > 0
-z
Reject H0 if Z < -Za
z
Reject H0 if Z > Za
Statistics for Managers Using Microsoft Excel, 5e
H0: 1 2 = 0
H1: 1 2 0
/2
/2
-z/2
z/2
Reject H0 if Z < -Za/2
or Z > Za/2
Chap 10-11
Two-Sample Tests
Independent Populations
Independent
Population Means
Assumptions:
Samples are randomly and
independently drawn
1 and 2 known
Populations are normally
distributed
1 and 2 unknown
Population variances are
unknown but assumed equal
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-12
Two-Sample Tests
Independent Populations
Independent
Population Means
1 and 2 known
1 and 2 unknown
Forming interval estimates:
The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate
the test statistic is a t value
with (n1 + n2 2) degrees
of freedom
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-13
Two-Sample Tests
Independent Populations
Independent
Population Means
1 and 2 known
The pooled standard
deviation is:
( n1 1)S1 + ( n 2 1)S2
2
1 and 2 unknown
Sp =
Statistics for Managers Using Microsoft Excel, 5e
(n1 1) + ( n 2 1)
Chap 10-14
2
Two-Sample Tests
Independent Populations
Independent
Population Means
The test statistic is:
( X X )(
t=
1
1
2 )
1 1
S +
n n
2
1
2
p
1 and 2 known
1 and 2 unknown
2
Where t has (n1 + n2 2) d.f., and
( n1 1) S12 + ( n2 1) S2 2
S2 =
p
Statistics for Managers Using Microsoft Excel, 5e
(n1 1) + (n2 1)
Chap 10-15
Two-Sample Tests
Independent Populations
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 NASDAQ
Number
21
25
Sample mean
3.27
2.53
Sample std dev 1.30
1.16
Assuming both populations are approximately normal with equal
variances, is there a difference in average yield ( = 0.05)?
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-16
Two-Sample Tests
Independent Populations
The test statistic is:
(X
t=
1
)
X 2 ( 1 2 )
1 1
S +
n n
2
1
2
p
=
( 3.27 2.53 ) 0
1
1
1.5021 +
21 25
= 2.040
( n1 1)S12 + ( n2 1)S2 2 = ( 21 1)1.30 2 + ( 25 1)1.16 2
S2 =
p
(n1 1) + (n2 1)
(21 - 1) + (25 1)
Statistics for Managers Using Microsoft Excel, 5e
= 1.5021
Chap 10-17
Two-Sample Tests
Independent Populations
H0: 1 - 2 = 0 i.e. (1 = 2)
Reject H0
Reject H0
H1: 1 - 2 0 i.e. (1 2)
= 0.05
.025
df = 21 + 25 - 2 = 44
-2.0154
Critical Values: t = 2.0154
Test Statistic: 2.040
.025
0 2.0154
t
2.040
Decision: Reject H0 at = 0.05
Conclusion: There is evidence
of a difference in the means.
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-18
Independent Populations
Unequal Variance
If you cannot assume population variances
are equal, the pooled-variance t test is
inappropriate
Instead, use a separate-variance t test, which
includes the two separate sample variances in
the computation of the test statistic
The computations are complicated and are
best performed using Excel
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-19
Two-Sample Tests
Independent Populations
Independent
Population Means
1 and 2 known
The confidence interval for
1 2 is:
(
)
2
2
1 2
X1 X 2 Z
+
n1 n 2
1 and 2 unknown
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-20
Two-Sample Tests
Independent Populations
Independent
Population Means
The confidence interval for
1 2 is:
( X X ) t
1
2
1 and 2 known
n1 + n 2 - 2
1 1
S +
n n
2
1
2
p
Where
1 and 2 unknown
( n1 1) S12 + ( n2 1) S2 2
S2 =
p
Statistics for Managers Using Microsoft Excel, 5e
(n1 1) + (n2 1)
Chap 10-21
Two-Sample Tests
Related Populations
Tests Means of 2 Related Populations
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
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-22
Two-Sample Tests
Related Populations
The ith paired difference is Di , where
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, D, is known.
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-23
Two-Sample Tests
Related Populations
The test statistic for the mean difference is a Z
value:
D D
Z=
D
n
Where
D = hypothesized mean difference
D = population standard deviation of differences
n = the sample size (number of pairs)
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-24
Two-Sample Tests
Related Populations
If D is unknown, you can estimate the
unknown population standard deviation with a
sample standard deviation:
n
SD =
(Di D) 2
i =1
n 1
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-25
Two-Sample Tests
Related Populations
The test statistic for D is now a t statistic:
D D
t=
SD
n
n
Where t has n - 1 d.f.
and SD is:
SD =
Statistics for Managers Using Microsoft Excel, 5e
(D
i=1
i
D)
2
n1
Chap 10-26
Two-Sample Tests
Related Populations
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: D 0
H 1 : D < 0
H0: D 0
H 1 : D > 0
H0: D = 0
H1: D 0
-t
Reject H0 if t < -ta
t
Reject H0 if t > ta
Statistics for Managers Using Microsoft Excel, 5e
/2
/2
-t/2
t/2
Reject H0 if t < -ta/2
or t > ta/2
Chap 10-27
Two-Sample Tests
Related Populations Example
Assume you send your salespeople to a customer
service training workshop. Has the training made a
difference the in number of complaints? You collect the
following data:
Salesperson
Number of Complaints
Before (1)
After (2)
Difference, Di
(2-1)
C.B.
6
4
-2
T.F.
20
6
-14
M.H.
3
2
-1
R.K.
0
0
0
M.O
4
0
-4
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-28
Two-Sample Tests
Related Populations Example
Salesperson
Number of Complaints
Before (1)
Difference, Di
(2-1)
After (2)
C.B.
6
4
-2
T.F.
20
6
-14
M.H.
3
2
-1
R.K.
0
0
0
4
0
M.O
n
D=
Di
i =1
n
= 4.2
SD =
(D
i
D)
-4
2
n 1
= 5.67
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-29
Two-Sample Tests
Related Populations Example
Has the training made a difference in the number of
complaints (at the = 0.01 level)?
H0: D = 0
H1: D 0
Critical Value = 4.604
d.f. = n - 1 = 4
Test Statistic:
t=
D D 4.2 0
=
= 1.66
SD / n 5.67/ 5
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-30
Two-Sample Tests
Related Populations Example
Reject
Reject
/2
/2
- 4.604
4.604
- 1.66
Decision: Do not reject H0
(t statistic is not in the reject
region)
Conclusion: There is no
evidence of a significant change
in the number of complaints
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-31
Two-Sample Tests
Related Populations
The confidence interval for D ( known) is:
D
DZ
n
Where
n = the sample size (number of pairs in the paired sample)
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-32
Two-Sample Tests
Related Populations
The confidence interval for D ( unknown) is:
SD
D t n1
n
n
where
SD =
(Di D)2
Statistics for Managers Using Microsoft Excel, 5e
i=1
n 1
Chap 10-33
Two Population Proportions
Goal: Test a hypothesis or form a confidence
interval for the difference between two
independent population proportions, 1 2
Assumptions:
n11 5 , n1(1-1) 5
n22 5 , n2(1-2) 5
The point estimate for the difference is p1 - p2
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-34
Two Population Proportions
Since you begin by assuming the null
hypothesis is true, you assume 1 = 2 and pool
the two sample (p) estimates.
The pooled estimate for
the overall proportion is:
X1 + X 2
p=
n1 + n 2
where X1 and X2 are the number of
successes in samples 1 and 2
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-35
Two Population Proportions
The test statistic for p1 p2 is a Z statistic:
Z=
where
(
p1 p2 ) ( 1 2 )
1 1
p (1 p) +
n n
2
1
p=
X1 + X 2
X
X
, P = 1 , P2 = 2
1
n1 + n 2
n1
n2
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-36
Two Population Proportions
Hypothesis for Population Proportions
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: 1 2
H 1 : 1 < 2
H0: 1 2
H1: 1 > 2
H0: 1 = 2
H 1: 1 2
i.e.,
i.e.,
i.e.,
H0: 1 2 0
H 1 : 1 2 < 0
H0: 1 2 0
H 1: 1 2 > 0
H0: 1 2 = 0
H 1 : 1 2 0
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-37
Two Population Proportions
Hypothesis for Population Proportions
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: 1 2 0
H 1 : 1 2 < 0
H0: 1 2 0
H 1: 1 2 > 0
H0: 1 2 = 0
H 1 : 1 2 0
-z
Reject H0 if Z < -Z
z
Reject H0 if Z > Z
Statistics for Managers Using Microsoft Excel, 5e
/2
/2
-z/2
z/2
Reject H0 if Z < -Z/2
or Z
> Z/2
Chap 10-38
Two Independent Population
Proportions: Example
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 of 72 men, 36 indicated
they would vote Yes and, in a sample of 50
women, 31 indicated they would vote Yes
Test at the .05 level of significance
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-39
Two Independent Population
Proportions: Example
H0: 1 2 = 0 (the two proportions are equal)
H1: 1 2 0 (there is a significant difference
between proportions)
The sample proportions are:
Men:
p1 = 36/72 = .50
Women:
p2 = 31/50 = .62
The pooled estimate for the overall proportion is:
p=
X1 + X 2 36 + 31 67
=
=
= .549
n1 + n 2 72 + 50 122
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-40
Two Independent Population
Proportions: Example
Reject H0
.025
The test statistic for 1 2 is:
z=
=
(
p1 p2 ) ( 1 2 )
1 1
p (1 p) +
n n
2
1
( .50 .62) ( 0)
1
1
.549 (1 .549) +
72 50
.025
-1.96
-1.31
= 1.31
Critical Values = 1.96
For = .05
Reject H0
1.96
Decision: Do not reject H0
Conclusion: There is no evidence of a
significant difference in proportions who
will vote yes between men and women.
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-41
Two Independent Population
Proportions
The confidence interval for 1 2 is:
(
p1 p2 ) Z
p1 (1 p1 ) p2 (1 p2 )
+
n1
n2
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-42
Testing Population Variances
Purpose: To determine if two independent
populations have the same variability.
H0: 12 = 22
H1: 12 22
H0: 12 22
H1: 12 < 22
H0: 12 22
H1: 12 > 22
Two-tail test
Lower-tail test
Upper-tail test
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-43
Testing Population Variances
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
n2 - 1 = denominator degrees of freedom
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-44
Testing Population Variances
The F critical value is found from the F table
There are two appropriate degrees of
freedom: numerator and denominator.
In the F table,
numerator degrees of freedom determine the
column
denominator degrees of freedom determine the
row
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-45
Testing Population Variances
Lower-tail test
Upper-tail test
H0: 12 22
H1: 12 < 22
H0: 12 22
H1: 12 > 22
0
0
Reject
H0
FL
Do not
reject H0
Reject H0 if F < FL
Statistics for Managers Using Microsoft Excel, 5e
Do not
reject H0
FU
Reject H0
Reject H0 if F > FU
Chap 10-46
Testing Population Variances
Two-tail test
H0: 12 = 22
H1: 12 22
/2
/2
0
FL
Do not
reject H0
FU
Reject H0
F
rejection region
for a two-tail test is:
Statistics for Managers Using Microsoft Excel, 5e
2
S1
F = 2 > FU
S2
2
S1
F = 2 < FL
S2
Chap 10-47
Testing Population Variances
To find the critical F values:
1. Find FU from the F table for n1 1
numerator and n2 1 denominator degrees
of freedom.
2. Find FL using the formula:
FL =
1
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)
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-48
Testing Population Variances
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 & NASDAQ at the = 0.05 level?
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-49
Testing Population Variances
Form the hypothesis test:
H0: 21 22 = 0
(there is no difference between variances)
H1: 21 22 0
(there is a difference between variances)
FU:
FL:
Numerator:
n1 1 = 21 1 = 20 d.f.
Denominator:
n2 1 = 25 1 = 24 d.f.
Numerator:
n2 1 = 25 1 = 24 d.f.
Denominator:
n1 1 = 21 1 = 20 d.f.
FU = F.025, 20, 24 = 2.33
Statistics for Managers Using Microsoft Excel, 5e
FL = 1/F.025, 24, 20 = 0.41
Chap 10-50
Testing Population Variances
The test statistic is:
S12 1.30 2
F= 2=
= 1.256
2
S 2 1.16
/2 = .025
/2 = .025
0
Reject H0
Do not
reject H0
FL=0.41
F = 1.256 is not in the
rejection region, so we do
not reject H0
Reject H0
FU=2.33
Conclusion: There is insufficient
evidence of a difference in
variances at = .05
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-51
F
Chapter Summary
In this chapter, we have
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
Performed separate-variance t test
Statistics for Managers Using Microsoft Excel, 5e
Chap 10-52
Chapter Summary
In this chapter, we have
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, 5e
Chap 10-53