Statistics for
Business and Economics
6th Edition
Chapter 11
Hypothesis Testing II
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-1
Chapter Goals
After completing this chapter, you should be able to:

Test hypotheses for the difference between two population means

Two means, matched pairs

Independent populations, population variances known

Independent populations, population variances unknown but
equal

Complete a hypothesis test for the difference between two
proportions (large samples)

Use the chi-square distribution for tests of the variance of a normal
distribution

Use the F table to find critical F values

Complete an F test for the equality of two variances
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-2
Two Sample Tests
Two Sample Tests
Population
Means,
Matched
Pairs
Population
Means,
Independent
Samples
Population
Proportions
Population
Variances
Examples:
Same group
before vs. after
treatment
Group 1 vs.
independent
Group 2
Proportion 1 vs.
Proportion 2
Variance 1 vs.
Variance 2
(Note similarities to Chapter 9)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-3
Matched Pairs
Tests Means of 2 Related Populations
Matched
Pairs



Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
di = xi - yi

Assumptions:
 Both Populations Are Normally Distributed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-4
Test Statistic: Matched Pairs
Matched
Pairs
The test statistic for the mean
difference is a t value, with
n – 1 degrees of freedom:
d  D0
t
sd
n
Where
D0 = hypothesized mean difference
sd = sample standard dev. of differences
n = the sample size (number of pairs)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-5
Decision Rules: Matched Pairs
Paired Samples
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μx – μy  0
H1: μx – μy < 0
H0: μx – μy ≤ 0
H1: μx – μy > 0
H0: μx – μy = 0
H1: μx – μy ≠ 0
a
a
-ta
ta
Reject H0 if t < -tn-1, a
Where
Reject H0 if t > tn-1, a
t
d  D0
sd
n
a/2
-ta/2
a/2
ta/2
Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
has n - 1 d.f.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-6
Matched Pairs Example
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:
di

d = n
Number of Complaints:
(2) - (1)

Salesperson
C.B.
T.F.
M.H.
R.K.
M.O.
Before (1)
After (2)
Difference, di
6
20
3
0
4
4
6
2
0
0
- 2
-14
- 1
0
- 4
-21
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
= - 4.2
Sd 
2
(d

d
)
 i
n 1
 5.67
Chap 11-7
Matched Pairs: Solution
 Has the training made a difference in the number of
complaints (at the a = 0.01 level)?
H0: μx – μy = 0
H1: μx – μy  0
a = .01
d = - 4.2
Critical Value = ± 4.604
d.f. = n - 1 = 4
Reject
Reject
a/2
a/2
- 4.604
4.604
- 1.66
Decision: Do not reject H0
(t stat is not in the reject region)
Test Statistic:
d  D0  4.2  0
t

 1.66
sd / n 5.67/ 5
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Conclusion: There is not a
significant change in the
number of complaints.
Chap 11-8
Difference Between Two Means
Population means,
independent
samples

Goal: Form a confidence interval
for the difference between two
population means, μx – μy
Different data sources
 Unrelated
 Independent

Sample selected from one population has no effect on the
sample selected from the other population
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-9
Difference Between Two Means
(continued)
Population means,
independent
samples
σx2 and σy2 known
Test statistic is a z value
σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
Test statistic is a a value from the
Student’s t distribution
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-10
σx2 and σy2 Known
Population means,
independent
samples
σx2 and σy2 known
σx2 and σy2 unknown
Assumptions:
*
 Samples are randomly and
independently drawn
 both population distributions
are normal
 Population variances are
known
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-11
σx2 and σy2 Known
(continued)
When σx2 and σy2 are known and
both populations are normal, the
variance of X – Y is
Population means,
independent
samples
2
σx2 and σy2 known
σx2 and σy2 unknown
σ 2X Y
*
2
σy
σx


nx
ny
…and the random variable
Z
(x  y)  (μX  μY )
2
σ 2x σ y

nX nY
has a standard normal distribution
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-12
Test Statistic,
σx2 and σy2 Known
Population means,
independent
samples
σx2 and σy2 known
The test statistic for
μx – μy is:
*
σx2 and σy2 unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

x  y   D0
z
2
σy
σx

nx
ny
2
Chap 11-13
Hypothesis Tests for
Two Population Means
Two Population Means, Independent Samples
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μx  μy
H1: μx < μy
H0: μx ≤ μy
H1: μx > μy
H0: μx = μy
H1: μx ≠ μy
i.e.,
i.e.,
i.e.,
H0: μx – μy  0
H1: μx – μy < 0
H0: μx – μy ≤ 0
H1: μx – μy > 0
H0: μx – μy = 0
H1: μx – μy ≠ 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-14
Decision Rules
Two Population Means, Independent
Samples, Variances Known
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μx – μy  0
H1: μx – μy < 0
H0: μx – μy ≤ 0
H1: μx – μy > 0
H0: μx – μy = 0
H1: μx – μy ≠ 0
a
a
-za
Reject H0 if z < -za
za
Reject H0 if z > za
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
a/2
-za/2
a/2
za/2
Reject H0 if z < -za/2
or z > za/2
Chap 11-15
σx2 and σy2 Unknown,
Assumed Equal
Assumptions:
Population means,
independent
samples
 Samples are randomly and
independently drawn
σx2 and σy2 known
 Populations are normally
distributed
σx2 and σy2 unknown
σx2 and σy2
assumed equal
*
 Population variances are
unknown but assumed equal
σx2 and σy2
assumed unequal
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-16
σx2 and σy2 Unknown,
Assumed Equal
(continued)
Forming interval
estimates:
Population means,
independent
samples
 The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
*
 use a t value with
(nx + ny – 2) degrees of
freedom
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-17
Test Statistic,
σx2 and σy2 Unknown, Equal
The test statistic for
μx – μy is:
σx2 and σy2 unknown
σx2 and σy2
assumed equal
*
t
σx2 and σy2
assumed unequal
 x  y    μx  μy 
1 1 
S   
n n 
y 
 x
2
p
Where t has (n1 + n2 – 2) d.f.,
and
sp2 
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(n x  1)s 2x  (n y  1)s 2y
nx  ny  2
Chap 11-18
σx2 and σy2 Unknown,
Assumed Unequal
Assumptions:
Population means,
independent
samples
 Samples are randomly and
independently drawn
σx2 and σy2 known
 Populations are normally
distributed
σx2 and σy2 unknown
 Population variances are
unknown and assumed
unequal
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
*
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-19
σx2 and σy2 Unknown,
Assumed Unequal
(continued)
Forming interval estimates:
Population means,
independent
samples
 The population variances are
assumed unequal, so a pooled
variance is not appropriate
σx2 and σy2 known
 use a t value with  degrees
of freedom, where
σx2 and σy2 unknown
2
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
*
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
 s2x
s2y 
( )  ( )
n y 
 n x
v
2
2
2
2


sy
 sx 

 /(n y  1)
  /(n x  1) 


n
y
 nx 
 
Chap 11-20
Test Statistic,
σx2 and σy2 Unknown, Unequal
σx2 and σy2 unknown
The test statistic for
μx – μy is:
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
*
t
(x  y)  D0
Where t has  degrees of freedom:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
σ
σ

nX nY
2
y
2
x
v
 s2x
s2y 
( )  ( )
n y 
 n x
2
2
 s2 
 s2x 
  /(n x  1)   y  /(n y  1)
n 
 nx 
 y
2
Chap 11-21
Decision Rules
Two Population Means, Independent
Samples, Variances Unknown
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μx – μy  0
H1: μx – μy < 0
H0: μx – μy ≤ 0
H1: μx – μy > 0
H0: μx – μy = 0
H1: μx – μy ≠ 0
a
a
-ta
Reject H0 if t < -tn-1, a
ta
Reject H0 if t > tn-1, a
a/2
-ta/2
a/2
ta/2
Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
Where t has n - 1 d.f.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-22
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
Assuming both populations are
approximately normal with
equal variances, is
there a difference in average
yield (a = 0.05)?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-23
Calculating the Test Statistic
The test statistic is:

X  X   μ  μ 
t

1
2
1
2
1 1
S   
 n1 n2 
2
p
3.27  2.53   0
1 
 1
1.5021 

 21 25 
2
2
2
2








n

1
S

n

1
S
21

1
1.30

25

1
1.16
1
2
2
S2  1

p
(n1  1)  (n2  1)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
(21 - 1)  (25  1)
 2.040
 1.5021
Chap 11-24
Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
a = 0.05
df = 21 + 25 - 2 = 44
Critical Values: t = ± 2.0154
Reject H0
.025
-2.0154
Reject H0
.025
0 2.0154
t
2.040
Test Statistic:
Decision:
3.27  2.53
t
 2.040 Reject H0 at a = 0.05
1 
 1
Conclusion:
1.5021  

 21 25 
There is evidence of a
difference in means.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-25
Two Population Proportions
Population
proportions
Goal: Test hypotheses for the
difference between two population
proportions, Px – Py
Assumptions:
Both sample sizes are large,
nP(1 – P) > 9
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-26
Two Population Proportions
(continued)
Population
proportions

The random variable
Z
(pˆ x  pˆ y )  (p x  p y )
pˆ x (1 pˆ x ) pˆ y (1 pˆ y )

nx
ny
is approximately normally distributed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-27
Test Statistic for
Two Population Proportions
The test statistic for
H0: Px – Py = 0
is a z value:
Population
proportions
z
 pˆ
x
 pˆ y 
pˆ 0 (1  pˆ 0 ) pˆ 0 (1  pˆ 0 )

nx
ny
Where
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
pˆ 0 
nxpˆ x  nypˆ y
nx  ny
Chap 11-28
Decision Rules: Proportions
Population proportions
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: px – py  0
H1: px – py < 0
H0: px – py ≤ 0
H1: px – py > 0
H0: px – py = 0
H1: px – py ≠ 0
a
a
-za
Reject H0 if z < -za
za
Reject H0 if z > za
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
a/2
-za/2
a/2
za/2
Reject H0 if z < -za/2
or z > za/2
Chap 11-29
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 Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-30
Example:
Two Population Proportions
(continued)

The hypothesis test is:
H0: PM – PW = 0 (the two proportions are equal)
H1: PM – PW ≠ 0 (there is a significant difference between
proportions)

The sample proportions are:
 Men:
p̂M = 36/72 = .50
p̂ W = 31/50 = .62
 Women:
 The estimate for the common overall proportion is:
pˆ 0 
nxpˆ x  nypˆ y
nx  ny
72(36/72)  50(31/50) 67


 .549
72  50
122
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-31
Example:
Two Population Proportions
(continued)
The test statistic for PM – PW = 0 is:
z

 pˆ
 pˆ W 
pˆ 0 (1  pˆ 0 ) pˆ 0 (1  pˆ 0 )

n1
n2
Reject H0
Reject H0
.025
.025
M
-1.96
-1.31
1.96
 .50  .62 
 .549 (1  .549) .549 (1  .549)  Decision: Do not reject H0



72
50

 Conclusion: There is not
  1.31
Critical Values = ±1.96
For a = .05
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
significant evidence of a
difference between men
and women in proportions
who will vote yes.
Chap 11-32
Hypothesis Tests of
one Population Variance
Population
Variance
 Goal: Test hypotheses about the
population variance, σ2
 If the population is normally distributed,

2
n1
(n  1)s

σ2
2
follows a chi-square distribution with
(n – 1) degrees of freedom
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-33
Confidence Intervals for the
Population Variance
(continued)
Population
Variance
The test statistic for
hypothesis tests about one
population variance is
χ
2
n 1
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(n  1)s

σ 02
2
Chap 11-34
Decision Rules: Variance
Population variance
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: σ2  σ02
H1: σ2 < σ02
H0: σ2 ≤ σ02
H1: σ2 > σ02
H0: σ2 = σ02
H1: σ2 ≠ σ02
a
a
χ n21,a
χn21,1a
Reject H0 if
χ
2
n 1
χ
2
n 1,1a
Reject H0 if
χ n21  χn21,a
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a/2
a/2
χ n21,1a / 2
χn21,a / 2
Reject H0 if
χ n21  χ n21,a / 2
or
χn21  χn21,1a / 2
Chap 11-35
Hypothesis Tests for Two Variances
Tests for Two
Population
Variances
F test statistic
 Goal: Test hypotheses about two
population variances
H0: σx2  σy2
H1: σx2 < σy2
H0: σx2 ≤ σy2
H1: σx2 > σy2
H0: σx2 = σy2
H1: σx2 ≠ σy2
Lower-tail test
Upper-tail test
Two-tail test
The two populations are assumed to be
independent and normally distributed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-36
Hypothesis Tests for Two Variances
(continued)
Tests for Two
Population
Variances
F test statistic
The random variable
2
x
2
y
s /σ
F
s /σ
2
x
2
y
Has an F distribution with (nx – 1)
numerator degrees of freedom and
(ny – 1) denominator degrees of
freedom
Denote an F value with 1 numerator and 2
denominator degrees of freedom by
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-37
Test Statistic
Tests for Two
Population
Variances
The critical value for a hypothesis test
about two population variances is
s
F
s
F test statistic
2
x
2
y
where F has (nx – 1) numerator
degrees of freedom and (ny – 1)
denominator degrees of freedom
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-38
Decision Rules: Two Variances
Use sx2 to denote the larger variance.
H0: σx2 = σy2
H1: σx2 ≠ σy2
H0: σx2 ≤ σy2
H1: σx2 > σy2
a/2
a
0
Do not
reject H0
Reject H0
Fnx 1,ny 1,α
F
Reject H0 if F  Fnx 1,ny 1,α
0
Do not
reject H0
F
Reject H0
Fnx 1,ny 1,α / 2
rejection region for a twotail test is:

Reject H0 if F  Fnx 1,ny 1,α / 2
where sx2 is the larger of
the two sample variances
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-39
Example: F Test
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 a = 0.10 level?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-40
F Test: Example Solution


Form the hypothesis test:
H0: σx2 = σy2 (there is no difference between variances)
H1: σx2 ≠ σy2 (there is a difference between variances)
Find the F critical values for a = .10/2:
Degrees of Freedom:
 Numerator
(NYSE has the larger
standard deviation):


nx – 1 = 21 – 1 = 20 d.f.
Fnx 1, ny 1, α / 2
 F20 , 24 , 0.10/2  2.03
Denominator:

ny – 1 = 25 – 1 = 24 d.f.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-41
F Test: Example Solution
(continued)

The test statistic is:
H0: σx2 = σy2
H1: σx2 ≠ σy2
s 2x 1.30 2
F 2 
 1.256
2
s y 1.16


F = 1.256 is not in the rejection
region, so we do not reject H0
a/2 = .05
Do not
reject H0
Reject H0
F
F20 , 24 , 0.10/2  2.03
Conclusion: There is not sufficient evidence
of a difference in variances at a = .10
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-42
Two-Sample Tests in EXCEL
For paired samples (t test):

Tools | data analysis… | t-test: paired two sample for means
For independent samples:

Independent sample Z test with variances known:

Tools | data analysis | z-test: two sample for means
For variances…
 F test for two variances:

Tools | data analysis | F-test: two sample for variances
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-43
Two-Sample Tests in PHStat
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-44
Sample PHStat Output
Input
Output
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-45
Sample PHStat Output
(continued)
Input
Output
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-46
Chapter Summary

Compared two dependent samples (paired
samples)


Compared two independent samples



Performed paired sample t test for the mean
difference
Performed z test for the differences in two means
Performed pooled variance t test for the differences
in two means
Compared two population proportions

Performed z-test for two population proportions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-47
Chapter Summary
(continued)

Used the chi-square test for a single population
variance

Performed F tests for the difference between
two population variances

Used the F table to find F critical values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 11-48