Airlines.xlsx DATADestinationSouthwest Fare ($)US Airways Fare ($)Chicago706706Nashville718674Denver980980Dallas10781284Atlanta654654Orlando820748Tampa820728Fort Lauderdale832822Phoenix11141918Las Vegas11381918 BrandZTechFin.xlsx DATABrandBrand Value 2014 ($mil)Brand Value Change (%)RegionProduct SectorApple24699267North AmericaTechnologyGoogle1736529North AmericaTechnologyMicrosoft11550028North AmericaTechnologyIBM93987-13North AmericaTechnologyVisa9196216North AmericaFinancial InstitutionsTencent/QQ7657243AsiaTechnologyFacebook7112199North AmericaTechnologyWells Fargo593109North AmericaFinancial InstitutionsMastercard401882North AmericaFinancial InstitutionsBaidu4004135AsiaTechnologyICBC Asia38808-8AsiaFinancial InstitutionsSAP382255Continental EuropeTechnologyAmerican Express3809311North AmericaFinancial InstitutionsHSBC24029-11United KingdomFinancial InstitutionsRBC239896North AmericaFinancial Institutionshp2303918North AmericaTechnologyChina Construction Bank22065-12AsiaFinancial InstitutionsOracle216804North AmericaTechnologySamsung21602-17AsiaTechnologyTD206383North AmericaFinancial InstitutionsCommonwealth Bank20599-2AsiaFinancial InstitutionsAgricultural Bank of China2018911AsiaFinancial InstitutionsAccenture2018311North AmericaTechnologyIntel1838558North AmericaTechnologyANZ17702-7AsiaFinancial InstitutionsCiti174861North AmericaFinancial InstitutionsBank of China1643816AsiaFinancial InstitutionsCisco1606017North AmericaTechnologySiemens15496-8Continental EuropeTechnologyHuawei15335ERROR:#N/AAsiaTechnologyUS Bank14786-1North AmericaFinancial InstitutionsJP Morgan135229North AmericaFinancial InstitutionsWestpac 124206AsiaFinancial InstitutionsLinkedIn12200-2North AmericaTechnologySantander1218110Continental EuropeFinancial InstitutionsChase116610North AmericaFinancial InstitutionsING1156018Continental EuropeFinancial InstitutionsTwitter11447-17North AmericaTechnologyBank of America1133512North AmericaFinancial InstitutionsScotiabank11044-3North AmericaFinancial Institutions lsxl8e_ppt_ch10.ppt Copyright © 2017, 2014, 2011 Pearson Education, Inc. Chapter 10 - * Two-Sample Tests Chapter 10 Copyright © 2017, 2014, 2011 Pearson Education, Inc. Chapter 10 - * Objectives In this chapter, you learn: How to compare the means of two independent populations. How to compare the means of two related populations. How to compare the proportions of two independent populations. How to compare the variances of two independent populations. Copyright © 2017, 2014, 2011 Pearson Education, Inc. Chapter 10 - * Online TopicEffect Size: Section 10.5. Copyright © 2017, 2014, 2011 Pearson Education, Inc. Chapter 10 - * Two-Sample Tests Two-Sample Tests Population Means, Independent Samples Population Means, Related Samples Population Variances Group 1 vs. Group ...

1. Airlines.xlsx
DATADestinationSouthwest Fare ($)US Airways Fare
($)Chicago706706Nashville718674Denver980980Dallas1078128
4Atlanta654654Orlando820748Tampa820728Fort
Lauderdale832822Phoenix11141918Las Vegas11381918
BrandZTechFin.xlsx
DATABrandBrand Value 2014 ($mil)Brand Value Change
(%)RegionProduct SectorApple24699267North
AmericaTechnologyGoogle1736529North
AmericaTechnologyMicrosoft11550028North
AmericaTechnologyIBM93987-13North
AmericaTechnologyVisa9196216North AmericaFinancial
InstitutionsTencent/QQ7657243AsiaTechnologyFacebook71121
99North AmericaTechnologyWells Fargo593109North
AmericaFinancial InstitutionsMastercard401882North
AmericaFinancial
InstitutionsBaidu4004135AsiaTechnologyICBC Asia38808-
8AsiaFinancial InstitutionsSAP382255Continental
EuropeTechnologyAmerican Express3809311North
AmericaFinancial InstitutionsHSBC24029-11United
KingdomFinancial InstitutionsRBC239896North
AmericaFinancial Institutionshp2303918North
AmericaTechnologyChina Construction Bank22065-
12AsiaFinancial InstitutionsOracle216804North
AmericaTechnologySamsung21602-
17AsiaTechnologyTD206383North AmericaFinancial
InstitutionsCommonwealth Bank20599-2AsiaFinancial
InstitutionsAgricultural Bank of China2018911AsiaFinancial
InstitutionsAccenture2018311North
AmericaTechnologyIntel1838558North
AmericaTechnologyANZ17702-7AsiaFinancial
InstitutionsCiti174861North AmericaFinancial InstitutionsBank

2. of China1643816AsiaFinancial InstitutionsCisco1606017North
AmericaTechnologySiemens15496-8Continental
EuropeTechnologyHuawei15335ERROR:#N/AAsiaTechnologyU
S Bank14786-1North AmericaFinancial InstitutionsJP
Morgan135229North AmericaFinancial InstitutionsWestpac
124206AsiaFinancial InstitutionsLinkedIn12200-2North
AmericaTechnologySantander1218110Continental
EuropeFinancial InstitutionsChase116610North
AmericaFinancial InstitutionsING1156018Continental
EuropeFinancial InstitutionsTwitter11447-17North
AmericaTechnologyBank of America1133512North
AmericaFinancial InstitutionsScotiabank11044-3North
AmericaFinancial Institutions
lsxl8e_ppt_ch10.ppt
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Two-Sample Tests
Chapter 10
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Objectives
In this chapter, you learn: How to compare the means of two
independent populations.
How to compare the means of two related populations.
How to compare the proportions of two independent
populations.

3. How to compare the variances of two independent populations.
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Online TopicEffect Size: Section 10.5.
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Two-Sample Tests
Two-Sample Tests
Population Means, Independent Samples
Population Means, Related Samples
Population Variances
Group 1 vs. Group 2
Same group before vs. after treatment
Variance 1 vs.
Variance 2
Examples:
Population Proportions
Proportion 1 vs. Proportion 2
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.

4. Chapter 10 - *
Difference Between Two Means
Population means, independent samples
Goal: Test hypothesis or form a confidence interval for the
difference between two population means, μ1 – μ2.
The point estimate for the difference is
X1 – X2
*
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Difference Between Two Means: Independent Samples
Population means, independent samples
*
Use Sp to estimate unknown σ. Use a Pooled-Variance t test.
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
Use S1 and S2 to estimate unknown σ1 and σ2. Use a
Separate-variance t testDifferent data
sourcesUnrelated.Independent.Sample selected from one
population has no effect on the sample selected from the other
population.

5. DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Hypothesis Tests for
Two Population Means
Lower-tail test:
H1: μ1 < μ2
i.e.,
H0: μ1 –
H1: μ1 – μ2 < 0
Upper-tail test:
H0: μ1 ≤ μ2
H1: μ1 > μ2
i.e.,
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
Two-tail test:
H0: μ1 = μ2
H1: μ1 ≠ μ2

6. i.e.,
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0
Two Population Means, Independent Samples
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Two Population Means, Independent Samples
Lower-tail test:
H0: μ1 –
H1: μ1 – μ2 < 0
Upper-tail test:
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
Two-tail test:
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0
a
a/2
a/2
a
-ta

7. -ta/2
ta
ta/2
Reject H0 if tSTAT < -ta
Reject H0 if tSTAT > ta
Reject H0 if tSTAT < -ta/2
or tSTAT > ta/2
Hypothesis tests for μ1 – μ2
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Population means, independent samples
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and
assumed equal
Assumptions:
Samples are randomly and
independently drawn.
Populations are normally
distributed or both sample
sizes are at least 30.
Population variances are
unknown but assumed equal.
*
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
DCOVA

8. Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Population means, independent samples
The pooled variance is:
The test statistic is:
Where tSTAT has d.f. = (n1 + n2 – 2).
(continued)
*
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and
assumed equal
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *

9. Population means, independent samples
The confidence interval for
μ1 – μ2 is:
Where tα/2 has d.f. = n1 + n2 – 2.
*
Confidence interval for µ1 - µ2 with σ1 and σ2 unknown and
assumed equal
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
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

10. 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
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Pooled-Variance t Test Example: Calculating the Test Statistic
The test statistic is:
(continued)
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)

11. df = 21 + 25 - 2 = 44
Critical Values: t = ± 2.0154
Test Statistic:
Pooled-Variance t Test Example: Hypothesis Test
Solution
Decision:
Conclusion:
Reject H0 at a = 0.05.
There is evidence of a difference in means.
t
0
2.0154
-2.0154
.025
Reject H0
Reject H0
.025
2.040

12. DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Pooled Variance t Test In Excel
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Since we rejected H0 can we be 95% confident that µNYSE >
µNASDAQ?
95% Confidence Interval for µNYSE - µNASDAQ:

13. Since 0 is less than the entire interval, we can be 95% confident
that µNYSE > µNASDAQ.
Pooled-Variance t Test Example: Confidence Interval for µ1 -
µ2
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Pooled Variance t Confidence Interval In Excel
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *

14. Population means, independent samples
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown, not
assumed equal
Assumptions:
Samples are randomly and
independently drawn.
Populations are normally
distributed or both sample
sizes are at least 30.
Population variances are
unknown and cannot be
assumed to be equal.
*
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.

15. Chapter 10 - *
Population means, independent samples
(continued)
*
σ1 and σ2 unknown, assumed equal
σ1 and σ2 unknown, not assumed equal
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and not
assumed equal
DCOVA
This test is known at the separate-variance t test.
The formulae for this test are not covered in this chapter. See
reference 3 for more details.
This test done in Excel is shown on the next slide.
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *

16. Separate-Variance t Test In Excel
(continued)
DCOVA
NYSE NASDAQ
Number 21 25
Sample mean 3.27 2.53
Sample std dev 1.30 1.16Using α=0.05 this test fails to
reject the null.
For this data whether we can assume equal
variances or not is important to determine
because when we assumed equal variances
the null was rejected.
In Section 10.4 a test to help determine whether
this is a reasonable assumption or not is discussed.
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Related Populations

17. The Paired Difference Test
Tests Means of 2 Related Populations
Paired or matched samples.
Repeated measures (before/after).
Use difference between paired values:
Eliminates Variation Among Subjects.Assumptions:Differences
are normally distributed.Or, if not Normal, use large samples.
Related samples
Di = X1i - X2i
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Related Populations
The Paired Difference Test
The ith paired difference is Di , where

18. Related samples
Di = X1i - X2i
The point estimate for the paired difference population mean
μD is D :
n is the number of pairs in the paired sample
The sample standard deviation is SD.
(continued)
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
The test statistic for μD is:
Paired samplesWhere tSTAT has n - 1 d.f.
The Paired Difference Test:
Finding tSTAT
DCOVA

19. Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Lower-tail test:
H1: μD < 0
Upper-tail test:
H0: μD ≤ 0
H1: μD > 0
Two-tail test:
H0: μD = 0
H1: μD ≠ 0
Paired Samples
The Paired Difference Test: Possible Hypotheses
a

20. a/2
a/2
a
-ta
-ta/2
ta
ta/2
Reject H0 if tSTAT < -ta
Reject H0 if tSTAT > ta
Reject H0 if tSTAT < -ta/2
or tSTAT > ta/2
Where tSTAT has n - 1 d.f.
DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
The confidence interval for μD is:
Paired samples
where
The Paired Difference Confidence Interval

21. DCOVA
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Assume 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:
Paired Difference Test: 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
DCOVA
D =

22. Di
n
= -4.2
Copyright © 2017, 2014, 2011
Pearson Education, Inc.
Chapter 10 - *
Has the training made a difference in the number of complaints
(at the 0.01 level)?
- 4.2
D =
H0: μD = 0
Test Statistic:
t0.005 = ± 4.604 d.f. = n - 1 = 4
Reject

23. - 4.604 4.604
Decision: Do not reject H0
(tstat is not in the rejection region).
Conclusion: There is insufficient evidence of a change in the
number of complaints.
Paired Difference Test: