Statistical Inference and Sampling

1
Statistical Inference and Sampling
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
2
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.

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
3
Estimation

• The population consists of every item of
  interest.
• The sample is randomly drawn from the population.
• Sample values should be selected randomly, one at
  a time, from the population.

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
4
Random Sampling and Estimation
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
Figure 7.1
5
Distribution of X

• The mean of the probability distribution for X
• Standard error of X standard deviation of the
  probability distribution for X

m
x
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
6
Distribution of X
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
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Figure 7.6
7
Distribution of X
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
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8
Probabilities in the Sampling Distribution of X
Figure 7.8
Introduction to Business Statistics, 5e
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9
Central Limit Theorem

• When obtaining large samples 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 m and a standard
  deviation (standard error) of

s
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
s

x
n
10
Central Limit Theorem
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
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Figure 7.10
11
Central Limit Theorem
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
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Figure 7.11
12
Central Limit Theorem
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
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Figure 7.12
13
Confidence for the Mean of a Normal Population (?
known)
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
Figure 7.16
14
Confidence for the Mean of a Normal Population (?
known)
P(-1.96 ? Z ? 1.96)
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
15
Confidence for the Mean of a Normal Population (?
known)
(1-?) 100 Confidence Interval
Introduction to Business Statistics, 5e
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16
Confidence for the Mean of a Normal Population (?
unknown)

• Students t Distribution
• Population variance unknown
• Degrees of freedom n - 1

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
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17
Students t Distribution
Introduction to Business Statistics, 5e
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18
Confidence for the Mean of a Normal Population (?
unknown)
X

m
t
s
/
n
Introduction to Business Statistics, 5e
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College Publishing
19
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
20
Selecting Necessary Sample SizeKnown ?

• 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
• Equation

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
21
Selecting Necessary Sample SizeKnown ?
æ
ö
?
E

Z
ç

a
/
2
n
è
ø
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
22
Selecting Necessary Sample SizeUnknown ?
2
Z

s
é
ù
a
/
2
n

ê
ú
E
ë
û
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
23
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

Introduction to Business Statistics, 5e
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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.

Introduction to Business Statistics, 5e
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25
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.

Introduction to Business Statistics, 5e
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26
Stratified Sampling

• Stratified sampling obtains more information due
  to the homogenous nature of each strata.
• Stratified sampling obtains a cross section fo
  the entire population.
• Obtain a mean within each strata as well as an
  estimate of ?.

Introduction to Business Statistics, 5e
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College Publishing
27
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

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing