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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-1
Chapter 9
Two-Sample Tests
Statistics for ManagersUsing Microsoft® Excel
4th Edition
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-2
Chapter Goals
After completing this chapter, you should be able to: Test hypotheses for the difference between two
independent population means (standard deviations known or unknown)
Test two means from related samples for the mean difference
Complete a Z test for the difference between two proportions
Use the F table to find critical F values
Complete an F test for the difference between two variances
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-3
Two Sample Tests
Two Sample Tests
Population Means,
Independent Samples
Means, Related Samples
Population Variances
Group 1 vs. independent Group 2
Same group before vs. after treatment
Variance 1 vs.Variance 2
Examples:
Population Proportions
Proportion 1 vs. Proportion 2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-4
Difference Between Two Means
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 unknown
Goal: Test hypotheses or form a confidence interval for the difference between two population means, μ1 – μ2
The point estimate for the difference is
X1 – X2
*
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-5
Independent Samples
Population means, independent
samples
Different data sources Unrelated 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 or pooled variance t test
*
σ1 and σ2 known
σ1 and σ2 unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-6
Difference Between Two Means
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 unknown
*
Use a Z test statistic
Use S to estimate unknown σ , use a t test statistic and pooled standard deviation
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-7
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 Known
Assumptions:
Samples are randomly and independently drawn
population distributions are normal or both sample sizes are 30
Population standard deviations are known
*σ1 and σ2 unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-8
Population means, independent
samples
σ1 and σ2 known …and the standard error of
X1 – X2 is
When σ1 and σ2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a Z-value…
2
22
1
21
XX n
σ
n
σσ
21
(continued)
σ1 and σ2 Known
*σ1 and σ2 unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-9
Population means, independent
samples
σ1 and σ2 known
2
22
1
21
2121
nσ
nσ
μμXXZ
The test statistic for
μ1 – μ2 is:
σ1 and σ2 Known
*σ1 and σ2 unknown
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-10
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-tail 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, 4e © 2004 Prentice-Hall, Inc. Chap 9-11
Two Population Means, Independent Samples
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 < -Z Reject H0 if Z > Z Reject H0 if Z < -Z/2
or Z > Z/2
Hypothesis tests for μ1 – μ2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-12
Population means, independent
samples
σ1 and σ2 known
2
22
1
21
21n
σ
n
σZXX
The confidence interval for
μ1 – μ2 is:
Confidence Interval, σ1 and σ2 Known
*σ1 and σ2 unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-13
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 Unknown
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
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-14
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 Unknown(continued)
*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
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-15
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 Unknown
The pooled standard deviation is
(continued)
1)n()1(n
S1nS1nS
21
222
211
p
*σ1 and σ2 unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-16
Population means, independent
samples
σ1 and σ2 known
σ1 and σ2 Unknown
Where t has (n1 + n2 – 2) d.f.,
and
21
2p
2121
n1
n1
S
μμXXt
The test statistic for
μ1 – μ2 is:
*σ1 and σ2 unknown
1)n()1(n
S1nS1nS
21
222
2112
p
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-17
Population means, independent
samples
σ1 and σ2 known
21
2p2-nn21
n
1
n
1StXX
21
The confidence interval for
μ1 – μ2 is:
Where
1)n()1(n
S1nS1nS
21
222
2112
p
*σ1 and σ2 unknown
Confidence Interval, σ1 and σ2 Unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-18
Pooled Sp 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
Assuming both populations are approximately normal with equal variances, isthere a difference in average yield ( = 0.05)?
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-19
Calculating the Test Statistic
1.5021
1)25(1)-(21
1.161251.30121
1)n()1(n
S1nS1nS
22
21
222
2112
p
2.040
251
211
5021.1
02.533.27
n1
n1
S
μμXXt
21
2p
2121
The test statistic is:
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-20
Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
= 0.05
df = 21 + 25 - 2 = 44Critical Values: t = ± 2.0154
Test Statistic: Decision:
Conclusion:
Reject H0 at = 0.05
There is evidence of a difference in means.
t0 2.0154-2.0154
.025
Reject H0 Reject H0
.025
2.040
2.040
251
211
5021.1
2.533.27t
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-21
Related Samples
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, if Not Normal, use large samples
Related samples
D = X1 - X2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-22
Mean Difference, σD Known
The ith paired difference is Di , whereRelated samples
Di = X1i - X2i
The point estimate for the population mean paired difference is D : n
DD
n
1ii
Suppose the population standard deviation of the difference scores, σD, is known
n is the number of pairs in the paired sample
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-23
The test statistic for the mean difference is a Z value:Paired
samples
n
σμD
ZD
D
Mean Difference, σD Known(continued)
WhereμD = hypothesized mean differenceσD = population standard dev. of differencesn = the sample size (number of pairs)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-24
Confidence Interval, σD Known
The confidence interval for D isPaired samples
n
σZD D
Where n = the sample size
(number of pairs in the paired sample)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-25
If σD is unknown, we can estimate the unknown population standard deviation with a sample standard deviation:
Related samples
1n
)D(DS
n
1i
2i
D
The sample standard deviation is
Mean Difference, σD Unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-26
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
SμD
tD
D
Where t has n - 1 d.f.
and SD is:
Mean Difference, σD Unknown(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-27
The confidence interval for D isPaired samples
1n
)D(DS
n
1i
2i
D
n
StD D
1n
where
Confidence Interval, σD Unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-28
Lower-tail test:
H0: μD 0H1: μD < 0
Upper-tail test:
H0: μD ≤ 0H1: μD > 0
Two-tail test:
H0: μD = 0H1: μD ≠ 0
Paired Samples
Hypothesis Testing for Mean Difference, σD Unknown
/2 /2
-t -t/2t t/2
Reject H0 if t < -t Reject H0 if t > t Reject H0 if t < -t
or t > t Where t has n - 1 d.f.
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-29
Assume you send your salespeople to a “customer service” training workshop. 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, 4e © 2004 Prentice-Hall, Inc. Chap 9-30
Has the training made a difference in the number of complaints (at the 0.01 level)?
- 4.2D =
1.6655.67/
04.2
n/S
μDt
D
D
H0: μD = 0H1: μD 0
Test Statistic:
Critical Value = ± 4.604 d.f. = n - 1 = 4
Reject
/2
- 4.604 4.604
Decision: Do not reject H0
(t stat is not in the reject region)
Conclusion: There is not a significant change in the number of complaints.
Paired Samples: Solution
Reject
/2
- 1.66 = .01
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-31
Two Population Proportions
Goal: test a hypothesis or form a confidence interval for the difference between two population proportions,
p1 – p2
The point estimate for the difference is
Population proportions
Assumptions: n1p1 5 , n1(1-p1) 5
n2p2 5 , n2(1-p2) 5
21 ss pp
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-32
Two Population Proportions
Population proportions
21
21
nn
XXp
The pooled estimate for the overall proportion is:
where X1 and X2 are the numbers from samples 1 and 2 with the characteristic of interest
Since we begin by assuming the null hypothesis is true, we assume p1 = p2
and pool the two ps estimates
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-33
Two Population Proportions
Population proportions
21
21ss
n1
n1
)p1(p
ppppZ 21
The test statistic for
p1 – p2 is a Z statistic:
(continued)
2
2s
1
1s
21
21
n
Xp ,
n
Xp ,
nn
XXp
21
where
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-34
Confidence Interval forTwo Population Proportions
Population proportions
2
ss
1
ssss n
)p(1p
n
)p(1pZpp 2211
21
The confidence interval for
p1 – p2 is:
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-35
Hypothesis Tests forTwo Population Proportions
Population proportions
Lower-tail test:
H0: p1 p2
H1: p1 < p2
i.e.,
H0: p1 – p2 0H1: p1 – p2 < 0
Upper-tail test:
H0: p1 ≤ p2
H1: p1 > p2
i.e.,
H0: p1 – p2 ≤ 0H1: p1 – p2 > 0
Two-tail test:
H0: p1 = p2
H1: p1 ≠ p2
i.e.,
H0: p1 – p2 = 0H1: p1 – p2 ≠ 0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-36
Hypothesis Tests forTwo Population Proportions
Population proportions
Lower-tail test:
H0: p1 – p2 0H1: p1 – p2 < 0
Upper-tail test:
H0: p1 – p2 ≤ 0H1: p1 – p2 > 0
Two-tail test:
H0: p1 – p2 = 0H1: p1 – p2 ≠ 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
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-37
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 at the .05 level of significance
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-38
The hypothesis test is:H0: p1 – p2 = 0 (the two proportions are equal)
H1: p1 – p2 ≠ 0 (there is a significant difference between proportions)
The sample proportions are: Men: ps1 = 36/72 = .50
Women: ps2 = 31/50 = .62
.549122
67
5072
3136
nn
XXp
21
21
The pooled estimate for the overall proportion is:
Example: Two population Proportions
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-39
The test statistic for p1 – p2 is:
Example: Two population Proportions
(continued)
.025
-1.96 1.96
.025
-1.31
Decision: Do not reject H0
Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women.
1.31
501
721
.549)(1.549
0.62.50
n1
n1
)p(1p
ppppz
21
21ss 21
Reject H0 Reject H0
Critical Values = ±1.96For = .05
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-40
Hypothesis Tests for Variances
Tests for TwoPopulation Variances
F test statistic
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2Two-tail test
Lower-tail test
Upper-tail test
H0: σ12 σ2
2
H1: σ12 < σ2
2
H0: σ12 ≤ σ2
2
H1: σ12 > σ2
2
*
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-41
Hypothesis Tests for Variances
Tests for TwoPopulation Variances
F test statistic22
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
*
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-42
The F critical value is found from the F table
The 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
The F Distribution
where df1 = n1 – 1 ; df2 = n2 – 122
21
S
SF
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-43
F 0
Finding the Rejection Region
L22
21
U22
21
FS
SF
FS
SF
rejection region for a two-tail test is:
FL Reject H0
Do not reject H0
F 0
FU Reject H0Do not
reject H0
F 0 /2
Reject H0Do not reject H0 FU
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2
H0: σ12 σ2
2
H1: σ12 < σ2
2
H0: σ12 ≤ σ2
2
H1: σ12 > σ2
2
FL
/2
Reject H0
Reject H0 if F < FL
Reject H0 if F > FU
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-44
Finding the Rejection Region
F 0 /2
Reject H0Do not reject H0 FU
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2
FL
/2
Reject H0
(continued)
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)
*UL F
1F 1. Find FU from the F table
for n1 – 1 numerator and
n2 – 1 denominator
degrees of freedom
To find the critical F values:
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-45
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, 4e © 2004 Prentice-Hall, Inc. Chap 9-46
F Test: Example Solution
Form the hypothesis test:
H0: σ21 – σ2
2 = 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
Find the F critical values for = .05:
Numerator: n2 – 1 = 25 – 1 = 24 d.f.
Denominator: n1 – 1 = 21 – 1 = 20 d.f.
FL = 1/F.025, 24, 20 = 1/2.41
= .41
FU: FL:
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-47
The test statistic is:
0
256.116.1
30.1
S
SF
2
2
22
21
/2 = .025
FU=2.33Reject H0Do not
reject H0
H0: σ12 = σ2
2
H1: σ12 ≠ σ2
2
F Test: Example Solution
F = 1.256 is not in the rejection region, so we do not reject H0
(continued)
Conclusion: There is not sufficient evidence of a difference in variances at = .05
FL=0.41
/2 = .025
Reject H0
F
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-48
Two-Sample Tests in EXCEL
For independent samples: Independent sample Z test with variances known:
Tools | data analysis | z-test: two sample for means
For paired samples (t test): Tools | data analysis… | t-test: paired two sample for means
For variances… F test for two variances:
Tools | data analysis | F-test: two sample for variances
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-49
Two-Sample Tests in PHStat
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-50
Sample PHStat Output
Input
Output
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-51
Sample PHStat Output
Input
Output
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-52
Chapter Summary
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
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-53
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
(continued)