Statistics for business economics 7th by paul newbold chapter 08

Statistics for
Business and Economics
7th Edition

Chapter 8
Estimation: Additional Topics

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

Ch. 8-1


Chapter Goals
After completing this chapter, you should be able to:


Form confidence intervals for the difference between
two means from dependent samples



Form confidence intervals for the difference between
two independent population means (standard deviations
known or unknown)



Compute confidence interval limits for the difference
between two independent population proportions



Determine the required sample size to estimate a mean
or proportion within a specified margin of error

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

Ch. 8-2


Estimation: Additional Topics
Chapter Topics
Confidence Intervals
Population
Means,
Dependent
Samples

Population
Means,
Independent
Samples

Population
Proportions

Examples:
Same group
before vs. after
treatment

Group 1 vs.
independent
Group 2

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Proportion 1 vs.
Proportion 2

Sample Size
Determination

Large
Populations

Finite
Populations

Ch. 8-3


8.1

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



Eliminates Variation Among Subjects
Assumptions:
 Both Populations Are Normally Distributed

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Ch. 8-4


Mean Difference
The ith paired difference is di , where
Dependent
samples

di = x i - y i
The point estimate for
the population mean
paired difference is d :
The sample

standard
deviation is:

n

d=

∑d
i =1

i

n

n

Sd =

2
(d
−
d
)
∑ i
i=1

n −1

n is the number of matched pairs in the sample
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Ch. 8-5


Confidence Interval for
Mean Difference
Dependent
samples

The confidence interval for difference
between population means, μd , is

d − t n−1,α/2

Sd
Sd
< μd < d + t n−1,α/2
n
n

Where
n = the sample size
(number of matched pairs in the paired sample)

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Ch. 8-6


Confidence Interval for

Mean Difference
(continued)
Dependent
samples



The margin of error is

ME = t n−1,α/2



sd
n

tn-1,α/2 is the value from the Student’s t
distribution with (n – 1) degrees of freedom
for which
α
P(t n−1 > t n−1,α/2 ) =
2

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Ch. 8-7


Paired Samples Example
Six people sign up for a

weight loss program. You
collect the following data:


Dependent
samples

Person
1
2
3
4
5
6

Weight:
Before (x)
After (y)
136
205
157
138
175
166

125
195
150
140
165

160

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Difference, di
11
10
7
-2
10
6
42

Σ di

d = n
= 7.0
Sd =

2
(d
−
d
)
∑ i

n −1

= 4.82
Ch. 8-8

Paired Samples Example
(continued)
Dependent
samples



For a 95% confidence level, the appropriate t value is
tn-1,α/2 = t5,.025 = 2.571



The 95% confidence interval for the difference between
means, μd , is

d − t n−1,α/2
7 − (2.571)

Sd
S
< μd < d + t n−1,α/2 d
n
n

4.82
4.82
< μd < 7 + (2.571)
6

6

− 1.94 < μd < 12.06
Since this interval contains zero, we cannot be 95% confident, given this
limited data, that the weight loss program helps people lose weight
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Ch. 8-9


Difference Between Two Means:
Independent Samples

8.2

Population means,
independent
samples


Different data sources
 Unrelated
 Independent




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

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

The point estimate is the difference between the two
sample means:

x–y

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Ch. 8-10


Difference Between Two Means:
Independent Samples
(continued)

Population means,
independent
samples
σx2 and σy2 known

Confidence interval uses zα/2

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

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Confidence interval uses a value
from the Student’s t distribution
Ch. 8-11


σx2 and σy2 Known
Population means,
independent
samples
σx2 and σy2 known

Assumptions:

*

σx2 and σy2 unknown

 Samples are randomly and
independently drawn
 both population distributions
are normal
 Population variances are
known

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Ch. 8-12

σx2 and σy2 Known
(continued)

When σx and σy 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. 8-13


Confidence Interval,
σx2 and σy2 Known
Population means,
independent
samples
σx2 and σy2 known

*

The confidence interval for
μx – μy is:

σx2 and σy2 unknown

(x − y) − z α/2

σ 2X σ 2Y
σ 2X σ 2Y
+
< μX − μY < (x − y) + z α/2
+
nx ny
nx ny

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Ch. 8-14


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
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Ch. 8-15


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
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 use a t value with
(nx + ny – 2) degrees of
freedom
Ch. 8-16


σx2 and σy2 Unknown,
Assumed Equal
(continued)

Population means,
independent
samples

The pooled variance is

σx and σ known

2

2
y

σx2 and σy2 unknown
σx2 and σy2
assumed equal

*

sp2 =

(n x − 1)s 2x + (n y − 1)s2y
nx + ny − 2

σx2 and σy2
assumed unequal
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Ch. 8-17


Confidence Interval,
σx2 and σy2 Unknown, Equal
σx2 and σy2 unknown
σx2 and σy2
assumed equal

*

The confidence interval for
μ1 – μ2 is:

sp2

sp2

σx2 and σy2
assumed unequal

(x − y) − t nx +n y −2,α/2

Where

sp2 =

sp2
nx

+

ny

< μX − μY < (x − y) + t nx +n y −2,α/2

nx

+

sp2
ny

(n x − 1)s 2x + (n y − 1)s2y
nx + ny − 2

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Ch. 8-18


Pooled Variance Example
You are testing two computer processors for speed.
Form a confidence interval for the difference in CPU
speed. You collect the following speed data (in Mhz):
CPUx
Number Tested
17
Sample mean
3004
Sample std dev
74

CPUy
14
2538
56

Assume both populations are
normal with equal variances,

and use 95% confidence
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Ch. 8-19


Calculating the Pooled Variance
The pooled variance is:
2
2
(
)
(
)
n
−
1
S
+
n
−
1
S
(
17 − 1) 74 2 + (14 − 1) 56 2
x
x
y
y
2

S =
=
p

(n x − 1) + (n y − 1)

(17 - 1) + (14 − 1)

= 4427.03

The t value for a 95% confidence interval is:
t n x +ny −2 , α/2 = t 29 , 0.025 = 2.045

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Ch. 8-20


Calculating the Confidence Limits


The 95% confidence interval is

(x − y) − t n x +n y −2,α/2

(3004 − 2538) − (2.054)

sp2
nx

+

sp2
ny

< μX − μY < (x − y) + t n x +ny −2,α/2

sp2
nx

+

sp2
ny

4427.03 4427.03
4427.03 4427.03
+
< μX − μY < (3004 − 2538) + (2.054)
+
17
14
17
14

416.69 < μX − μY < 515.31
We are 95% confident that the mean difference in
CPU speed is between 416.69 and 515.31 Mhz.
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Ch. 8-21


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

*

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Ch. 8-22


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 ν degrees
of freedom, where

σx2 and σy2 unknown

2

σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal

*

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 s 2x
s 2y 
( ) + ( )
n y 
 n x
v=
2
2
2
2


s
 sx 
  /(n x − 1) +  y  /(n y − 1)
n 
 nx 
 y
Ch. 8-23

Confidence Interval,
σx2 and σy2 Unknown, Unequal
σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal

(x − y) − tν ,α/2

*

The confidence interval for
μ1 – μ2 is:

2
2
s 2x s y
s2x s y
+
< μX − μY < (x − y) + tν ,α/2
+
nx ny
nx ny

Where
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v=

 s 2x
s 2y 
( ) + ( )
n y 
 n x
2

2

2

 s2 
 s 2x 
  /(n x − 1) +  y  /(n y − 1)
n 
 nx 
 y
Ch. 8-24


8.3

Population
proportions

Two Population Proportions
Goal: Form a confidence interval for
the difference between two
population proportions, Px – Py
Assumptions:

Both sample sizes are large (generally at
least 40 observations in each sample)
The point estimate for
the difference is

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pˆ x − pˆ y
Ch. 8-25