Statistics for business economics 7th by paul newbold chapter 10

Statistics for
Business and Economics
7th Edition

Chapter 10
Hypothesis Testing:
Additional Topics
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Ch. 10-1


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

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Ch. 10-2


Two Sample Tests
Two Sample Tests

Population
Means,
Dependent
Samples

Population
Means,
Independent
Samples

Population
Proportions

Population
Variances

Examples:
Same group
before vs. after
treatment

Group 1 vs.
independent
Group 2

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Proportion 1 vs.
Proportion 2

Variance 1 vs.
Variance 2

Ch. 10-3

10.1

Dependent Samples
Tests Means of 2 Related Populations

Dependent
Samples





Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:

di = x i - y i


Assumptions:
 Both Populations Are Normally Distributed

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Ch. 10-4


Test Statistic:
Dependent Samples

The test statistic for the mean difference is a t value, with
n – 1 degrees of freedom:

Dependent
Samples

where

d  D0
t
sd
n

d

d
n

i

x  y

D0 = hypothesized mean difference
sd = sample standard dev. of differences
n = the sample size (number of pairs)
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Ch. 10-5

Decision Rules: Matched Pairs
Matched or 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




-t

t

Reject H0 if t < -tn-1, 
Where

Reject H0 if t > tn-1, 
t

d  D0
sd
n

 /2
-t/2

 /2
t/2

Reject H0 if t < -tn-1 , 

 or t
> tn-1 , 

has n - 1 d.f.

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Ch. 10-6


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)
6
20
3
0
4

After (2)
4
6
2
0
0

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Difference, di
- 2
-14
- 1

0
- 4
-21

= - 4.2
Sd 

2
(d

d
)
 i

n 1

 5.67
Ch. 10-7


Matched Pairs: Solution
 Has the training made a difference in the number of

complaints (at the  = 0.05 level)?
H0: μx – μy = 0
H1: μx – μy 
0
 = .05
d = - 4.2
Critical Value = ± 2.776

d.f. = n − 1 = 4

Test Statistic:

d  D0  4.2  0
t

 1.66
sd / n 5.67/ 5
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Reject

Reject

/2

/2

- 2.776

2.776

- 1.66

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.
Ch. 10-8


10.2

Difference Between Two Means

Population means,
independent
samples


Goal: Form a confidence interval
for the difference between two
population means, μx – μy

Different populations
 Unrelated
 Independent




Sample selected from one population has no effect on the
sample selected from the other population

Normally distributed

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Ch. 10-9


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
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Test statistic is a a value from the
Student’s t distribution
Ch. 10-10


σx2 and σy2 Known
Population means,
independent
samples

σx2 and σy2 known

Assumptions:

*



Samples are randomly and
independently drawn



both population distributions
are normal



Population variances are
known

σx2 and σy2 unknown

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Ch. 10-11


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
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Ch. 10-12


Test Statistic,
σx2 and σy2 Known
H0 : μx  μy D0

Population means,
independent
samples
σx2 and σy2 known

*

σx2 and σy2 unknown

The test statistic for
μx – μy is:


x  y   D0
z

2

2

σy
σx

nx
ny

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Ch. 10-13


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

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Ch. 10-14


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




-z

Reject H0 if z < -z

z
Reject H0 if z > z

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 /2
-z/2

 /2
z/2

Reject H0 if z < -z/2

or z

> z/2
Ch. 10-15


σx2 and σy2 Unknown,
Assumed Equal
Assumptions:

Population means,
independent
samples
σx2 and σy2 known



Samples are randomly and
independently drawn



Populations are normally
distributed



Population variances are
unknown but assumed equal

σx2 and σy2 unknown
σx2 and σy2
assumed equal

*

σx2 and σy2
assumed unequal
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Ch. 10-16


σx2 and σy2 Unknown,
Assumed Equal
(continued)

Population means,
independent
samples



The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ



use a t value with
(nx + ny – 2) degrees of
freedom

σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2
assumed equal

*

σx2 and σy2
assumed unequal
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Ch. 10-17


Test Statistic,
σx2 and σy2 Unknown, Equal
The test statistic for
μx – μy is:

σx2 and σy2 unknown
σx2 and σy2
assumed equal

*

σx2 and σy2
assumed unequal

t

 x  y    μx  μy 
2
p

2
p

s
s

nx ny

Where t has (n1 + n2 – 2) d.f.,
and

sp2 
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(nx  1)s2x  (ny  1)s2y
nx  ny  2
Ch. 10-18


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




-t
Reject H0 if
t < -t (n1+n2 – 2), 

t
Reject H0 if
t > t (n1+n2 – 2), 

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

 /2

-t/2

 /2
t/2

Reject H0 if
t < -t (n1+n2 – 2), or
t > t (n1+n2 – 2), 

Ch. 10-19


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
Number
21
Sample mean
3.27
Sample std dev 1.30

NASDAQ
25
2.53
1.16

Assuming both populations are

approximately normal with
equal variances, is
there a difference in average
yield ( = 0.05)?
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Ch. 10-20


Calculating the Test Statistic
The test statistic is:


X
t

1



 X 2  μ1  μ2 
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)

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(21 - 1)  (25  1)

2.040

1.5021

Ch. 10-21


Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)

Reject H0

Reject H0

H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
 = 0.05
df = 21 + 25 − 2 = 44
Critical Values: t = ± 2.0154

Test Statistic:
3.27  2.53
t
 2.040
1 
 1
1.5021  

 21 25 

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.025

-2.0154

.025

0 2.0154

t

2.040

Decision:
Reject H0 at  = 0.05
Conclusion:
There is evidence of a
difference in means.
Ch. 10-22


σx2 and σy2 Unknown,
Assumed Unequal
Assumptions:

Population means,
independent
samples
σx2 and σy2 known



Samples are randomly and
independently drawn



Populations are normally
distributed



Population variances are
unknown and assumed
unequal

σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal

*

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Ch. 10-23


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



use a t value with  degrees
of freedom, where

σx2 and σy2 known
σx2 and σy2 unknown

2

σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal

*

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 s 2x
s 2y 
( )  ( )
ny 
 nx
v
2

2
2
2


s
 sx 
  /(nx  1)   y  /(ny  1)
n 
 nx 
 y
Ch. 10-24


Test Statistic,
σx2 and σy2 Unknown, Unequal
The test statistic for
μx – μy is:

σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal

*

t

(x  y)  D0

Where t has  degrees of freedom:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

2
y

s
s

nX nY
2
x

v

 s2x
s2y 
( )  ( )
ny 
 nx
2

2

2

 s2 
 s2x 
  /(nx  1)   y  /(ny  1)

n 
 nx 
 y
Ch. 10-25