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Stat 0302B
Business Statistics
Spring 20102011
Chapter IV
Population and Sample
§ 4.1
Introduction
Population
:
a group of individuals about which we wish to make an inference.
We usually do not gather information from the entire population.
Sample
:
a subgroup of the population. We usually have data on the sampled
individuals.
Random Sample
:
sample drawn randomly by a method involving unpredictable
components in such a way that each possible sample of size
n
have known chance to be selected, e.g. lucky draw on each
individual in the population.
Example 4.1
Farmer Jane owns 1,264 sheep. These sheep constitute her entire
population
of
sheep. If 15 sheep are selected to be sheared, then these 15 sheep represent a
sample
from Jane’s population. Further, if the 15 sheep were selected at
random
from these 1,264 sheep, then they would constitute a
random sample
.
Parameter
: A numerical characteristic of a population, such as the population
mean or the population standard deviation.
Statistic
:
Numerical characteristic of a sample. Statistics may be calculated
from data in a sample.
Statistical Inference
: A conclusion about a population based on sampled
observations.
Example 4.2
Suppose we want to investigate the weights of wool that can be sheared from the
sheep. Then the mean and standard deviation of the weights of wool of the entire
1,264 sheep are the
population parameters
. For a random sample of size 15 from
this population, the sample mean and sample standard deviation of the weights of
wool of these 15 sheep are the
statistics
.
Statistical inference
may involve the
estimation of the population parameters by these sample statistics.
p. 84
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Business Statistics
Spring 20102011
§ 4.2
Sampling Distribution of Sample Statistics
Sampling distribution
: the distribution of possible values of a sample statistic
over all random samples of a given size.
Example 4.3
Consider a hypothetical population with size
5
=
N
:
{ 2, 4, 6, 8, 10 }
If we draw a number randomly from this population and denote it as
, then the
probabilistic behaviour of
can be described by the following distribution:
1
X
1
X
()
(
)
5
1
10
8
6
4
2
1
1
1
1
1
=
=
=
=
=
=
=
=
=
=
X
P
X
P
X
P
X
P
X
P
This is called the
population distribution
. Moreover, it can be easily determined
that the population mean and population variance are
6
1
=
=
X
E
μ
,
( )
8
1
2
=
=
X
Var
σ
.
Suppose we draw one more number from the
same population
and denote it as
(sample with replacement). The distribution of
is the same as
and
are independent. Then
2
X
1
,
X
2
X
1
X
2
X
{ }
2
1
,
X
X
forms a random sample with size
2
=
n
.
From a particular random sample, we can calculate the following sample statistics:
2
2
1
X
X
X
+
=
,
( )
2
1
2
1
2
2
1
2
2
2
1
2
X
X
X
X
X
X
S
−
=
−
+
−
−
=
Since
are
random,
2
1
,
X
X
X
and
2
S
are also random. Their probability
distributions can be determined by the following table:
Sample
X
−
X
2
S
2
2
−
S
Probability
2, 2
2
 4
0
 8
1/25
2, 4
3
 3
2
 6
2/25
2, 6
4
 2
8
0
2/25
2, 8
5
 1
18
10
2/25
2, 10
6
0
32
24
2/25
4, 4
4
 2
0
 8
1/25
4, 6
5
 1
2
 6
2/25
4, 8
6
0
8
0
2/25
4, 10
7
1
18
10
2/25
6, 6
6
0
0
 8
1/25
p. 85
Stat 0302B
Business Statistics
Spring 20102011
6, 8
7
1
2
 6
2/25
6, 10
8
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This document was uploaded on 05/11/2011.
 Spring '11
 Statistics

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