Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-1 Statistics...
-
date post
18-Dec-2015 -
Category
Documents
-
view
217 -
download
0
Transcript of Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-1 Statistics...
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-1
Statistics for ManagersUsing Microsoft® Excel
5th Edition
Chapter 10
Two-Sample Tests
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-2
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 10-3
Two-Sample Tests Overview
Two Sample Tests
Independent Population
Means
Means, Related
Populations
Independent Population Variances
Group 1 vs. Group 2
Same group before vs. after treatment
Variance 1 vs.Variance 2
Examples
Independent Population Proportions
Proportion 1vs. Proportion 2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-4
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Different data sources 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 10-5
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 10-6
Testing Population Variances
Purpose: To determine if two independent populations have the same variability.
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2
H0: σ12 σ2
2
H1: σ12 < σ2
2
H0: σ12 ≤ σ2
2
H1: σ12 > σ2
2
Two-tail test Lower-tail test Upper-tail test
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-7
Testing Population Variances
22
21
S
SF
The F test statistic is:
= Variance of Sample 1
n1 - 1 = numerator degrees of freedom
n2 - 1 = denominator degrees of freedom
= Variance of Sample 2
21S
22S
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-8
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 NASDAQNumber 21 25Mean 3.27 2.53Std 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 10-9
F Test: Example Solution
The test statistic is:
256.116.1
30.1
S
SF
2
2
22
21
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2
The p-value is not < alpha, so we do not reject H0
(continued)
Conclusion: There is not sufficient evidence of a difference in the variances of dividend yield for the two exchanges
p-value= .588846 = .05
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-10
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 10-11
Hypothesis Tests forTwo Population Means
Lower tail test:
H0: μ1 μ2
H1: μ1 < μ2
i.e.,
H0: μ1 – μ2 0H1: μ1 – μ2 < 0
Upper tail test:
H0: μ1 ≤ μ2
H1: μ1 > μ2
i.e.,
H0: μ1 – μ2 ≤ 0H1: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 = μ2
H1: μ1 ≠ μ2
i.e.,
H0: μ1 – μ2 = 0H1: μ1 – μ2 ≠ 0
Two Population Means, Independent Samples
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-12
Two-Sample Tests for MeansIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Assumptions:
Samples are randomly and independently drawn
Populations are normally distributed
or large sample sizes are used
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-13
Difference Between Two Means
Population means, independent
samples
σ1 and σ2 unknownbut equal
σ1 and σ2 unknownbut unequal
Use s to estimate unknown σ Use pooled variance t test
where 1=2 or individual variance t test
where 12 (proven by F test)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-14
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
(continued)
1)n()1(n
S1nS1nS
21
222
211
p
σ1 and σ2 unknownbut equal
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-15
Population means, independent
samples
σ1 and σ2 Unknown but Equal
Where t has (n1 + n2 – 2) d.f.,
and
21
2p
21
n1
n1
S
XXt
The test statistic for
μ1 = μ2 is:
σ1 and σ2 unknownbut equal
1)n()1(n
S1nS1nS
21
222
2112
p
(continued)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-16
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 NASDAQNumber 21 25Sample mean 3.27 2.53Sample 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 10-17
Solution
H0: μ1 = μ2
H1: μ1 ≠ μ2
= 0.05
df = 21 + 25 - 2 = 44
Pooled Test Statistic:
p-value:
Decision:
Conclusion:
Reject H0 at = 0.05
There is sufficient evidence of a difference in mean dividend yield for the two exchanges.
t0
.025
Reject H0 Reject H0
.025
2.0397
251
211
5021.1
2.533.27t
.047407
F = 1.256p = .5888Do Not Reject σ1
2 = σ22
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-18
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 10-19
Independent PopulationsUnequal 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 10-20
The Unequal-Variance t Test
2
2
2
1
2
1
21 )(
ns
ns
XXt
1
)/(
1
)/(
)//(d.f.
2
22
22
1
21
21
22
221
21
n
ns
n
ns
nsns
),,(TDIST tailsdftp
Population means, independent
samples
σ1 and σ2 unknownand unequal
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-21
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 NASDAQNumber 21 25Sample mean 3.27 2.53Sample 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 10-22
Unequal Variance Solution
H0: 1 = 2
H1: 12
= 0.05
df = 92
Test Statistics:
Decision:
Conclusion:
Fail to Reject at = 0.05
There is not sufficient evidence of a difference in mean dividend yields for the two exchanges
t0
.025
Reject H0 Reject H0
.025
.1901
p-value =.1901t = 1.34
F = 3.9313p = .001794Reject HO: 21
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-23
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 10-24
Confidence IntervalsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
21
2p2-nn21
n
1
n
1SXX
21t
The confidence interval for
μ1 – μ2 is:
Where
1)n()1(n
S1nS1nS
21
222
2112
p
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-25
Two-Sample TestsRelated Populations
D = X1 - X2
Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values:
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 10-26
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.:
Paired samples
1n
)D(DS
n
1i
2i
D
n
SD
tD
Mean Difference, σD Unknown(continued)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-27
Assume you send your salespeople to a “customer service” training workshop to reduce complaints. Is the training effective? You collect the following data:
Paired Samples Example
Number of Complaints: (2) - (1)Salesperson Before (1) After (2) Difference, Di
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 -21
D = Di
n
5.67
1n
)D(DS
2i
D
= -4.2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-28
Has the training made a difference in the number of complaints (at the 0.05 level)?
- 4.2D =
1.6655.67/
4.2
n/St
D
D
H0: μD = 0H1: μD 0
Test Statistic:
d.f. = n - 1 = 4
Reject
/2
Decision: Do not reject H0
(p-value is not <.05)
Conclusion: There is not sufficient evidence of a change in the number of complaints.
Paired Samples: Solution
Reject
/2
-1.66 = .01
p-value= .1730
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-29
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 10-30
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 10-31
Two Population Proportions
Hypothesis for Population Proportions
Lower-tail test:
H0: π1 π2
H1: π1 < π2
i.e.,
H0: π1 – π2 0H1: π1 – π2 < 0
Upper-tail test:
H0: π1 ≤ π2
H1: π1 > π2
i.e.,
H0: π1 – π2 ≤ 0H1: π1 – π2 > 0
Two-tail test:
H0: π1 = π2
H1: π1 ≠ π2
i.e.,
H0: π1 – π2 = 0H1: π1 – π2 ≠ 0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-32
Two Population Proportions
Since you begin by assuming the null hypothesis is true, you assume π1 = π2 and pool
the two sample (p) estimates.
21
21
nn
XXp
The pooled estimate for the overall proportion is:
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 10-33
Two Population Proportions
21
21
11)1(
nnpp
ppZ
The test statistic for π1 = 2 is a Z statistic:
2
22
1
11
21
21
n
X ,
n
X ,
nn
XXp
PPwhere
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-34
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 10-35
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 10-36
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:
.549122
67
5072
3136
nn
XXp
21
21
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-37
Two Independent Population Proportions: Example
The test statistic for π1 = π2 is:
1.31
501
721
.549)(1.549
.62.50
n1
n1
)p(1p
z
21
21
pp
P-value for -1.31 is .1902At = .05
.025
-1.96 1.96
.025
-1.31
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.
Reject H0 Reject H0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-38
Confidence Interval - Two Independent Population Proportions
2
22
1
1121 n
)(1
n
)(1 ppppZpp
The confidence interval for π1 – π2 is:
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-39
Two Sample Tests in PHStat
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-40
Sample PHStat Output
Input
Output
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-41
Sample PHStat Output
Input
Output
(continued)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-42
Chapter SummaryIn 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 10-43
Chapter Summary
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
In this chapter, we have