Download Hypothesis Testing

Statistics for 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 with
equal variances
 The means of two independent populations with
unequal variances
 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 © 2008 Prentice-Hall, Inc.
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 © 2008 Prentice-Hall, Inc.
Variance 1 vs.
Variance 2
Chap 10-3
Two-Sample Tests
Independent Populations
 Different data sources
Independent
Population Means
σ1 and σ2 known
σ1 and σ2 unknown
 Independent: Sample
selected from one
population has no effect
on the sample selected
from the other population
 Use pooled variance t test,
or separate-variance t test
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-4
The Process
There are two tests to determine if the means of two
populations are different. One test assumes the
variances of the populations are equal the other is
needed if the variances are not equal. Rather than
make assumptions about the variances, I prefer to
perform an F test for the equality of population
variances and use the results to determine which
model should be used. Therefore the F test comes
first.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-5
Testing Population Variances
 Purpose: To determine if two independent
populations have the same variability.
H0: σ12 = σ22
H1: σ12 ≠ σ22
H0: σ12  σ22
H1: σ12 < σ22
H0: σ12 ≤ σ22
H1: σ12 > σ22
Two-tail test
Lower-tail test
Upper-tail test
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-6
Testing Population Variances
The F test statistic is:
2
1
2
2
S
F
S
S12 = Variance of Sample 1
n1 - 1 = numerator degrees of freedom
S22 = Variance of Sample 2
n2 - 1 = denominator degrees of freedom
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-7
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
& NASDAQ at the  = 0.05 level?
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-8
F Test: Example Solution
(continued)
H0: σ12 = σ22
H1: σ12 ≠ σ22
 The test statistic is:
S12 1.302
F 2 
 1.256
2
S2 1.16
 p-value= .588846
 = .05
 The p-value is not < alpha, so we do not reject H0
 Conclusion: There is not sufficient evidence of a difference in the
variances of dividend yield for the two exchanges
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-9
F test in PhStat
 PHStat | Two-Sample Tests | F Test for
Differences in Two Variances …
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-10
Hypothesis Tests for
Two Population Means
Two Population Means, Independent Samples
Lower tail test:
Upper tail test:
Two-tailed 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, 5e © 2008 Prentice-Hall, Inc.
Chap 10-11
Two-Sample Tests for Means
Independent Populations
Independent
Population Means
Assumptions:
 Samples are randomly and
independently drawn
σ1 and σ2 known
 Populations are normally
distributed
σ1 and σ2 unknown
 or large sample sizes are used
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-12
Difference Between Two Means
Population means,
independent
samples
σ1 and σ2 unknown
but equal
σ1 and σ2 unknown
but unequal
 Use s to estimate unknown σ
 Use pooled variance t test
where 1=2 or
 individual variance t test
where 12 (proven by F
test)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-13
σ1 and σ2 Unknown but Equal
(continued)
Population means,
independent
samples
σ1 and σ2 unknown
but equal
If an F test does not prove the
variances to be unequal, we
can assume that they are
equal and pool them. The
pooled standard deviation is
Sp 
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
n1  1S12  n2  1S2 2
(n1  1)  (n2  1)
Chap 10-14
σ1 and σ2 Unknown but Equal
(continued)
The test statistic for
μ1 = μ2 is:
Population means,
independent
samples
t
σ1 and σ2 unknown
but equal
X  X 
1
2
1 1 
S   
 n1 n 2 
2
p
Where t has (n1 + n2 – 2) d.f.,
and
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
S
2
p
2
2

n1  1S1  n2  1S2

(n1  1)  (n2  1)
Chap 10-15
Pooled Variance 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 NASDAQ
Number
21
25
Sample mean
3.27
2.53
Sample std dev 1.30
1.16
Is there a difference in the mean
dividend yield between the
NYSE & NASDAQ ( = 0.05)?
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-16
Solution
F = 1.256
p = .5888
H0: μ1 = μ2
Do Not Reject
H1: μ1 ≠ μ2
2 = σ 2
σ
1
2
 = 0.05
df = 21 + 25 - 2 = 44
Pooled Test Statistic:
t
3.27  2.53
1 1 
1.5021   
 21 25 
p-value:
 2.0397
.047407
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Reject H0
Reject H0
.025
.025
0
t
Decision:
Reject H0 at  = 0.05
Conclusion:
There is sufficient evidence of
a difference in mean dividend
yield for the two exchanges.
Chap 10-17
Pooled Variance t Test in
PhStat and Excel
 If the raw data is available
 Tools | Data Analysis | t-Test: Two Sample
Assuming Equal Variances
 If only summary statistics are available
 PHStat | Two-Sample Tests | t Test for
Differences in Two Means …
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
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 not in the Text but are
as follows:
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-19
The Unequal-Variance t Test
Population means,
independent
samples
σ1 and σ2 unknown
and unequal
(X 1  X 2)
t
2
2
s1 s2

n1 n2
( s / n1  s / n2 )
d.f. 
 ( s12 / n1 ) 2 ( s22 / n2 ) 2 



n2  1 
 n1  1
2
1
2
2
2
p  TDIST (t , df , tails )
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-20
Unequal Variance 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 NASDAQ
Number
21
25
Sample mean
3.27
2.53
Sample std dev 2.30
1.16
Is there a difference in the mean
dividend yield between the
NYSE & NASDAQ ( = 0.05)?
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-21
Unequal Variance Solution
Test Statistics:
H0: m1 = m2
H1: m1 m2
 = 0.05
df = 92
F = 3.9313
p = .001794
Reject HO:  1
 2
t = 1.34
p-value =.1901
Reject H0
Reject H0
.025
.025
0
.1901
t
Decision:
Fail to Reject at  = 0.05
Conclusion:
There is not sufficient evidence of a
difference in mean dividend yields
for the two exchanges
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-22
Unequal Variance t Test in
PhStat and Excel
 If the raw data is available
 Tools | Data Analysis | t-Test: Two Sample
Assuming Unequal Variances
 If only summary statistics are available
 Use the Excel spreadsheet downloaded from the
Homework web page
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-23
Confidence Intervals
Independent Populations
Independent
Population Means
The confidence interval for
μ1 – μ2 is:
X  X   t
1
2
σ1 and σ2 known
n1  n 2 - 2
1 1 
S   
 n1 n 2 
2
p
Where
σ1 and σ2 unknown
2
2




n

1
S

n

1
S
1
2
2
S2  1
p
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
(n1  1)  (n2  1)
Chap 10-24
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
or have large samples
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-25
Mean Difference, σD Unknown
(continued)
Paired
samples
If σD is unknown, we can estimate the unknown
population standard deviation with a sample
standard deviation. The test statistic for D is now
a t statistic, with n-1 d.f.:
D
t
SD
n
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
n
SD 
2
(D

D
)
 i
i1
n 1
Chap 10-26
Paired Samples Example
 Assume you send your salespeople to a “customer
service” training workshop to reduce complaints. 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, 5e © 2008 Prentice-Hall, Inc.
- 2
-14
- 1
0
- 4
-21
D =
 Di
n
= -4.2
SD 
2
(D

D
)
 i
n 1
 5.67
Chap 10-27
Paired Samples: Solution
 Has the training made a difference in the number of complaints
(at the 0.05 level)?
H0: μD = 0
H1: μD  0
 = .01
D = - 4.2
Reject
/2
/2
-1.66
d.f. = n - 1 = 4
Decision: Do not reject H0
Test Statistic:
D
 4.2
t

 1.66
SD / n 5.67/ 5
p-value=
Reject
.1730
(p-value is not <.05)
Conclusion: There is not
sufficient evidence of a change
in the number of complaints.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-28
Related Sample t Test in
PhStat and Excel
 If the raw data is available
 Tools | Data Analysis | t-Test: Paired Two Sample for Means
 If only summary statistics are available
 No test available
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-29
Two Population Proportions
Goal: Test a hypothesis or form a confidence
interval for two independent population
proportions, π1 = π2
Assumptions:
n1π1  5 , n1(1-π1)  5
n2π2  5 , n2(1-π2)  5
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-30
Two Population Proportions
Hypothesis for Population Proportions
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: π1  π2
H1: π1 < π2
H0: π1 ≤ π2
H1: π1 > π2
H0: π1 = π2
H1: π1 ≠ π2
i.e.,
i.e.,
i.e.,
H0: π1 – π2  0
H1: π1 – π2 < 0
H0: π1 – π2 ≤ 0
H1: π1 – π2 > 0
H0: π1 – π2 = 0
H1: π1 – π2 ≠ 0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-31
Two Population Proportions
Since you begin by assuming the null
hypothesis is true, you assume π1 = π2 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 © 2008 Prentice-Hall, Inc.
Chap 10-32
Two Population Proportions
The test statistic for π1 = 2 is a Z statistic:
Z 
where
 p1  p2 
1 1
p (1  p)   
 n1 n2 
p
X1  X 2
X
X
, P1  1 , P2  2
n1  n 2
n1
n2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-33
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 for a significant difference at the .05 level of
significance
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-34
Check the Assumptions
The sample proportions are:
 Men:
ps1 = 36/72 = .50
 Women: ps2 = 31/50 = .62
Assumptions
Men np=72(.5)=36 n(1-p)=72(.5)=36
Women np=50(.62)=31 n(1-p)50(.38)=19
All values > 5
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-35
Two Independent Population
Proportions: Example
 H0: π1 = π2 (the two proportions are equal)
 H1: π1  π2
 The sample proportions are:
 Men:
p1 = 36/72 = .50
 Women: p2 = 31/50 = .62
 The pooled estimate for the overall proportion is:
X1  X 2 36  31 67
p


 .549
n1  n 2 72  50 122
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-36
Two Independent Population
Proportions: Example
The test statistic for π1 = π2 is:
z

 p1  p2 
1 1 
p (1  p)   
 n1 n 2 
 .50  .62
1 
 1
.549 (1  .549)   
 72 50 
Reject H0
.025
.025
-1.96
-1.31
  1.31
P-value for -1.31 is .1902
At  = .05
Reject H0
1.96
Decision: Do not reject H0
There is not sufficient evidence of a
difference in the proportions of men
and women who will vote yes on
Proposition A.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-37
Confidence Interval - Two
Independent Population Proportions
The confidence interval for π1 – π2 is:
 p1  p2   Z
p1 (1  p1 ) p2 (1  p2 )

n1
n2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-38
Two Sample Tests in PHStat
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-39
Sample PHStat Output
Input
Output
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-40
Sample PHStat Output
(continued)
Input
Output
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-41
Chapter Summary
In this chapter, we have
Compared two independent samples
 Performed pooled variance t test for the differences
in two means
 Examined unequal 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 t test for the mean
difference
 Formed confidence intervals for the paired difference

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-42
Chapter Summary
In this chapter, we have
 Compared two population proportions
 Performed Z-test for equality of two population
proportions
 Formed confidence intervals for the difference
between two population proportions
 Performed F tests for the difference between two
population variances
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 10-43