statistics for managers using microsoft excel, 5e 2008 prentice-hall, inc.chap 10-1 statistics for...
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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 2TRANSCRIPT
![Page 1: Statistics for Managers Using Microsoft Excel, 5e 2008 Prentice-Hall, Inc.Chap 10-1 Statistics for Managers Using Microsoft Excel 5th Edition Chapter](https://reader036.vdocument.in/reader036/viewer/2022081514/5a4d1b8a7f8b9ab0599be41c/html5/thumbnails/1.jpg)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-1
Statistics for ManagersUsing Microsoft® Excel
5th Edition
Chapter 10Two-Sample Tests
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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 The means of two related populations The proportions of two independent
populations The variances of two independent
populations
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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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-4
Two-Sample Tests
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ1 – μ2
The point estimate for the difference between sample means:
X1 – X2
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-5
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 the difference between 2 sample means
Use Z test, pooled variance t test, or separate-variance t test
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-6
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Use a Z test statistic
Use S to estimate unknown σ, use a t test statistic
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-7
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Assumptions:
Samples are randomly and independently drawn
population distributions are normal
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-8
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
When σ1 and σ2 are known and both populations are normal, the test statistic is a Z-value and the standard error of X1 – X2 is
2
22
1
21
XX nσ
nσσ
21
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-9
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
2
22
1
21
2121
nσ
nσ
μμXXZ
The test statistic is:
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-10
Two-Sample TestsIndependent Populations
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-tail test:
H0: μ1 = μ2
H1: μ1 ≠ μ2
i.e.,
H0: μ1 – μ2 = 0H1: μ1 – μ2 ≠ 0
Two Independent Populations, Comparing Means
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-11
Two-Sample TestsIndependent PopulationsTwo Independent Populations, Comparing Means
Lower-tail test:
H0: μ1 – μ2 0H1: μ1 – μ2 < 0
Upper-tail test:
H0: μ1 – μ2 ≤ 0H1: μ1 – μ2 > 0
Two-tail test:
H0: μ1 – μ2 = 0H1: μ1 – μ2 ≠ 0
/2 /2
-z -z/2z z/2
Reject H0 if Z < -Za Reject H0 if Z > Za Reject H0 if Z < -Za/2
or Z > Za/2
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-12
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Assumptions: Samples are randomly and independently drawn
Populations are normally distributed
Population variances are unknown but assumed equal
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-13
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
Forming interval estimates:
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 with (n1 + n2 – 2) degrees of freedom
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-14
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
The pooled standard deviation is:
1)n()1(n
S1nS1nS21
222
211
p
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-15
Two-Sample TestsIndependent Populations
Where t has (n1 + n2 – 2) d.f., and
21
2p
2121
n1
n1S
μμXXt
The test statistic is:
1)n()1(n
S1nS1nS21
222
2112
p
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-16
Two-Sample TestsIndependent 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 NASDAQNumber 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16
Assuming both populations are approximately normal with equal variances, is there a difference in average yield ( = 0.05)?
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-17
Two-Sample TestsIndependent Populations
1.50211)25(1)-(21
1.161251.301211)n()1(n
S1nS1nS22
21
222
2112
p
2.040
251
2115021.1
02.533.27
n1
n1S
μμXXt
21
2p
2121
The test statistic is:
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-18
Two-Sample TestsIndependent Populations
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Test Statistic: 2.040
t0 2.0154-2.0154
.025
Reject H0 Reject H0
.025
Decision: Reject H0 at α = 0.052.040
Conclusion: There is evidence of a difference in the means.
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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 complicated and are best performed using Excel
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-20
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
2
22
1
21
21nσ
nσXX Z
The confidence interval for μ1 – μ2 is:
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-21
Two-Sample TestsIndependent Populations
Independent Population Means
σ1 and σ2 known
σ1 and σ2 unknown
21
2p2-nn21
n1
n1SXX
21t
The confidence interval for μ1 – μ2 is:
Where
1)n()1(n
S1nS1nS21
222
2112
p
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-22
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-23
Two-Sample TestsRelated PopulationsThe ith paired difference is Di , where
n
DD
n
1ii
Di = X1i - X2i
The point estimate for the population mean paired difference is D :
Suppose the population standard deviation ofthe difference scores, σD, is known.
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-24
Two-Sample TestsRelated PopulationsThe test statistic for the mean difference is a Z
value:
nσ
μDZD
D
WhereμD = hypothesized mean differenceσD = population standard deviation of differencesn = the sample size (number of pairs)
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-25
Two-Sample TestsRelated PopulationsIf σD is unknown, you can estimate the unknown population standard deviation with a sample standard deviation:
1n
)D(DS
n
1i
2i
D
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-26
Two-Sample TestsRelated Populations
1n
)D(DS
n
1i
2i
D
nS
μDtD
D
The test statistic for D is now a t statistic:
Where t has n - 1 d.f.
and SD is:
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-27
Two-Sample TestsRelated Populations
Lower-tail test:
H0: μD 0H1: μD < 0
Upper-tail test:
H0: μD ≤ 0H1: μD > 0
Two-tail test:
H0: μD = 0H1: μD ≠ 0
/2 /2
-t -t/2t t/2
Reject H0 if t < -ta Reject H0 if t > ta Reject H0 if t < -ta/2
or t > ta/2
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-28
Two-Sample TestsRelated Populations ExampleAssume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:
Salesperson Number of Complaints Difference, Di
(2-1)Before (1) After (2)
C.B. 6 4 -2T.F. 20 6 -14M.H. 3 2 -1R.K. 0 0 0M.O 4 0 -4
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-29
Two-Sample TestsRelated Populations Example
2.4n
DD
n
1ii
5.671n
)D(DS
2i
D
Salesperson Number of Complaints Difference, Di
(2-1)Before (1) After (2)
C.B. 6 4 -2T.F. 20 6 -14M.H. 3 2 -1R.K. 0 0 0M.O 4 0 -4
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-30
Two-Sample TestsRelated Populations ExampleHas the training made a difference in the number of complaints (at the α = 0.01 level)?
H0: μD = 0H1: μD 0
Critical Value = ± 4.604 d.f. = n - 1 = 4
Test Statistic:
1.6655.67/04.2
n/Sμt
D
D
D
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-31
Two-Sample TestsRelated Populations Example
Reject
- 4.604 4.604
Reject
/2
- 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
/2
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-32
Two-Sample TestsRelated Populations The confidence interval for μD (σ known) is:
nσDZD
Where n = the sample size (number of pairs in the paired sample)
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-33
Two-Sample TestsRelated Populations The confidence interval for μD (σ unknown) is:
1n
)D(DS
n
1i
2i
D
nStD D
1n
where
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-34
Two Population Proportions
Goal: Test a hypothesis or form a confidence interval for the difference between two independent population proportions, π1 – π2
Assumptions: n1π1 5 , n1(1-π1) 5
n2π2 5 , n2(1-π2) 5
The point estimate for the difference is p1 - p2
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-35
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
nnXXp
The pooled estimate for the overall proportion is:
where X1 and X2 are the number of successes in samples 1 and 2
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-36
Two Population Proportions
21
2121
11)1(nn
pp
ppZ
The test statistic for p1 – p2 is a Z statistic:
2
22
1
11
21
21
nX ,
nX ,
nnXXp
PPwhere
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-37
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-38
Two Population Proportions
Hypothesis for Population Proportions
Lower-tail test:
H0: π1 – π2 0H1: π1 – π2 < 0
Upper-tail test:
H0: π1 – π2 ≤ 0H1: π1 – π2 > 0
Two-tail test:
H0: π1 – π2 = 0H1: π1 – π2 ≠ 0
/2 /2
-z -z/2z z/2
Reject H0 if Z < -Z Reject H0 if Z > Z Reject H0 if Z < -Z
or Z > Z
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-39
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-40
Two Independent Population Proportions: Example H0: π1 – π2 = 0 (the two proportions are equal) H1: π1 – π2 ≠ 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:
.54912267
50723136
nnXXp
21
21
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-41
Two Independent Population Proportions: Example
The test statistic for π1 – π2 is:
1.31
501
721.549)(1.549
0.62.50
n1
n1)p(1p
z
21
2121
pp
Critical Values = ±1.96For = .05
.025
-1.96 1.96
.025
-1.31
Decision: Do not reject H0
Conclusion: There is no evidence of a significant difference in proportions who will vote yes between men and women.
Reject H0 Reject H0
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-42
Two Independent Population Proportions
2
22
1
1121 n
)(1n
)(1 ppppZpp
The confidence interval for π1 – π2 is:
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-43
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-44
Testing Population Variances
22
21
SSF
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-45
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-46
Testing Population Variances
0
FL Reject H0
Do not reject H0
H0: σ12 σ2
2
H1: σ12 < σ2
2
Reject H0 if F < FL
0
FU Reject H0Do not reject H0
H0: σ12 ≤ σ2
2
H1: σ12 > σ2
2
Reject H0 if F > FU
Lower-tail test Upper-tail test
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-47
Testing Population Variances
L22
21
U22
21
FSSF
FSSF
rejection region for a two-tail test is:F 0
/2
Reject H0Do not reject H0 FU
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2
FL
/2
Two-tail test
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-48
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.
*UL F
1F 2. Find FL using the formula:
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)
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-49
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 NASDAQNumber 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?
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-50
Testing Population Variances
Form the hypothesis test: H0: σ2
1 – σ22 = 0 (there is no difference between variances)
H1: σ21 – σ2
2 ≠ 0 (there is a difference between variances)
Numerator: n1 – 1 = 21 – 1 = 20 d.f.
Denominator: n2 – 1 = 25 – 1 = 24 d.f.
FU = F.025, 20, 24 = 2.33
Numerator: n2 – 1 = 25 – 1 = 24 d.f.
Denominator: n1 – 1 = 21 – 1 = 20 d.f.
FL = 1/F.025, 24, 20 = 0.41
FU: FL:
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-51
Testing Population Variances The test statistic is:
256.116.130.1
2
2
22
21
SSF
0 /2 = .025
FU=2.33Reject H0Do not
reject H0
FL=0.41
/2 = .025
Reject H0
F
F = 1.256 is not in the rejection region, so we do not reject H0
Conclusion: There is insufficient evidence of a difference in variances at = .05
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-52
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
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Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 10-53
Chapter Summary
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
In this chapter, we have