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Introduction to Business Statistics, 6e
Kvanli, Pavur, Keeling
Chapter 7 –
Statistical
Inference and
Sampling
Slides prepared by Jeff Heyl, Lincoln University
Thomson/South-Western Learning™
1
©2003 South-Western/Thomson
Simple Random Sampling
 All items in the population have the same
probability of being selected
 Finite Population: To be sure that a
simple random sample is obtained from a
finite population the items should be
numbered from 1 to N
 Nearly all statistical procedures require
that a random sample is obtained
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Estimation
 The population consists of every item of
interest
 Population mean is µ and is generally
not known
 The sample is randomly drawn from the
population
 Sample values should be selected
randomly, one at a time, from the
population
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Random Sampling and
Estimation
Population
(mean = µ)
X estimates µ
Figure 7.1
Sample
(mean = X)
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Distribution for Everglo
Bulb Lifetime
 = 50
|
|
300
350
|
µ = 400
|
|
450
500
X
Figure 7.2
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Sample Means
Figure 7.3
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Excel Histogram
Frequency Histogram
8
7
6
5
4
3
2
1
0
377 and
under 384
384 and
under 391
391 and
under 398
398 and
under 405
405 and
under 412
412 and
under 419
419 and
under 426
426 and
under 433
Class Limits
Figure 7.4
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Distribution of X
The mean of the probability distribution
for X = µX = µ
Standard error of X = standard deviation

of the probability distribution for X = X =
n
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Normal Curves
Population (mean = µ,
standard deviation = )
Random sample (mean = X,
standard deviation = s
 = 50
X = value from this
population
Assumes the individual
observations follow a
normal distribution
x =
x
50
10
X follows a normal distribution, centered
at µ with a standard deviation  / n
µx = 400
X
Figure 7.5
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Central Limit Theorem
When obtaining large samples (n > 30)
from any population, the sample mean X
will follow an approximate normal
distribution
What this means is that if you randomly
sample a large population the X
distribution will be approximately normal
with a mean µ and a standard deviation
(standard error) of

x =
n
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Distribution of X
Population
 = 50
µ = 400
x =
X
50
20
x =
50
10
= 11.18
= 15.81
X
µx = 400
(n = 10)
x =
50
50
x =
50
100
=5
= 7.07
µx = 400
(n = 50)
X
µx = 400
(n = 20)
X
µx = 400
(n = 100)
X
Figure 7.6
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Distribution of X
Mean = µx = µ
Standard deviation = x =
(standard error)

n
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Assembly Time
=3
Area = P(X > 22)
|
14
|
17
|
| |
µ = 20 22 23
|
26
X = assembly line
Figure 7.7
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Assembly Time
x = .77
Area = P(X > 22)
|
14
|
19.23
|
µx
|
20.77
|
22
X
Figure 7.8
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Assembly Time
Area = P(19 < X < 21)
|
19
|
µx = 20
|
21
X
Figure 7.9
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Central Limit Theorem
a+b
= 100
2
b-a
=
= 28.87
12
µ=
Uniform
population
a = 50
(n = 2)
X
µ = 100
(n = 5)
b = 150
X
X
X
µx
(n = 30)
By the CLT, µx = µ = 100
x =
28.87

=
= 5.27
30
n
Figure 7.10
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Central Limit Theorem
Exponential
population
µ = 100 = 
X
(n = 2)
X
(n = 5)
X
X
µx
(n = 30)
By the CLT, µx = µ = 100
x =

=
n
100
30
= 18.26
Figure 7.11
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Central Limit Theorem
U-shaped
population
|
µ
|
(n = 2)
X
X
|
(n = 5)
X
|
(n = 30)
X
Figure 7.12
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Sampling Without
Replacement
Mean = µx = µ
Standard deviation = x =
(standard error)

n
•
N-n
N-1
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Distribution of Sample Mean
N-n

N-1
n
8500
350 - 45
=
45
350 - 1
x =
= (1267.11)(.935)
= $1184.75
µx = 48,000
Observed value of X = $43,900
X = average
income of 45
female
managers
Figure 7.13
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Confidence Intervals
 known
µ=?
X
Figure 7.14
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Confidence Intervals
3
x =
= .6 minute
25
Area = P(X > 20) = .5
µx = 20
X = average
of 25
assembly
lines
Figure 7.15
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Confidence Intervals
Area = .475
-1.96
Area = .475
0
Z
1.96
Total area = .95
Figure 7.16
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Confidence for the Mean of a
Normal Population ( known)
Z=
X-µ
/ n
P(-1.96  Z  1.96) = .95
X-µ
P -1.96 ≤
≤ 1.96 = .95
/ n
P X - 1.96

n
≤ µ ≤ X + 1.96

n
= .95
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Confidence for the Mean of a
Normal Population ( known)
(1 - ) • 100% Confidence Interval
x - Z/2

, x + Z/2
n
E = margin of error = Z/2

n

n
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Confidence for the Mean of a
Normal Population ( known)
Area = .1
Area = .05
1.645
1.96
0
1.28
Area = .025
Z
Figure 7.17
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Excel Screens
Figure 7.18
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Excel Screens
Figure 7.19
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Excel Screens
Figure 7.20
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Confidence for the Mean of a
Normal Population ( unknown)
Student’s t Distribution
Population variance unknown
Degrees of freedom = n - 1
x - t/2, n - 1
s
n
to
x + t/2, n - 1
s
n
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Student’s t Distribution
Standard normal, Z
t curve with 20 df
t curve with 10 df
0
t
Figure 7.21
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Confidence Interval
Figure 7.22
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Confidence Interval
Figure 7.23
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Selecting Necessary Sample
Size Known 
 Sample size based on the level of
accuracy required for the application
 Maximum error: E
 Used to determine the necessary
sample size to provide the specified
level of accuracy
 Specified in advance
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Selecting Necessary Sample
Size Known 
E = Z/2

n
Z/2 • 
n=
E
2
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Selecting Necessary Sample
Size Unknown 
To obtain a rough approximation, ask
someone who is familiar with the data
to be collected:
1. What do you think will be the highest
value in the sample (H)?
2. What will be the lowest value (L)?
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Selecting Necessary Sample
Size Unknown 
H-L
 4
Z/2 • s
n=
E
2
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Other Sampling Procedures
 Population: the collection of all items
about which we are interested
 Sampling Unit: a collection of elements
selected from the population
 Cluster: a sampling unit that is a group
of elements from the population, such as
all adults in a particular city block
 Sampling frame: a list of population
elements
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Other Sampling Procedures
 Strata: are nonoverlapping
subpopulations
 Sampling design: specifies the
manner in which the sampling
units are to be selected
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Simple Random Sampling
Population mean: µ
Estimator:
∑x
X= n
Estimated standard error of X:

N-n
sx =
•
n
N-1
Approximate confidence interval:
X ± Z/2sx
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Systematic Sampling
 The sampling frame consists of N
records
 The sample of n is obtained by
sampling every kth record, where k is
an integer approximately equal N/n
 The sampling frame should be
ordered randomly
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Stratified Sampling
 Stratified sampling obtains more
information due to the homogenous
nature of each strata
 Stratified sampling obtains a cross
section of the entire population
 Obtain a mean within each strata as
well as an estimate of 
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Stratified Sampling
Use the following notation:
ni
Ni
N
n
Xi
si
= sample size in stratum i
= number of elements in stratum i
= total population size = ∑Ni
= total sample size = ∑ni
= sample mean in stratum i
= sample standard deviation in stratum i
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Stratified Sampling
Population mean: µ
Estimator:
∑NiXi
Xst = N
Estimated standard error of X:
sx =
st
Ni
∑
N
2
Ni - ni
Ni
si2
ni
Approximate confidence interval:
Xst ± Z/2sx
st
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Cluster Sampling
 Single-stage cluster sampling: randomly
select a set of clusters for sampling
Include all elements in the cluster in your
sample
 Two-stage cluster sampling: randomly
select a set of clusters for sampling
Randomly select elements from each
sampled cluster
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Cluster Sampling
Population mean: µ
Estimator:
∑Ti
Xc = ∑n
i
Estimated standard error of Xc:
sx =
c
M - m ∑(Ti - Xcni)2
m-1
mMN2
Approximate confidence interval:
Xc ± Z/2sx
c
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Confidence Interval
Constructing a Confidence Interval for a Population Mean
 known
 unknown
Use Table A-4 (Z)
Use Table A-5 (t)
Can use Table A-4 (Z) to
obtain approximate confidence
interval if n > 30
Figure 7.25
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