STA3032 Fall 2011, Chapter 8, Page 1 of 12
Chapter 8
Sampling Distribution of Sample Statistics
A parameter
is a function of population data. Parameters are numerical characteristics of
a population. They are fixed numbers, i.e., their
values do not change
from sample to
sample.
Some examples are μ, σ and
π.
A (sample) statistic
is a function of sample data. Statistics are numerical summaries of
sample data (that contain no unknown parameters). Their
values do change
from sample
to sample.
Some examples are
ˆ
ˆ
,
X S and
p
Statistics
are random variables since their values change from sample to sample. Like
any other random variable, we will talk about the
mean of a statistic
and standard
deviation of a statistic (called
standard error
of the statistic
) as well as the
distribution
of a statistic.
8.1 Sampling Distribution of a Sample Statistic
Sampling Distribution
of a statistic is the probability distribution of that statistic.
[At this point we need to use our imagination]
Suppose we decide to use simple random sampling to select
ALL POSSIBLE
SAMPLES
of some fixed size, n. (How many possible
SRS’s
of size n can you
select from a population of size N?)
After each sample of n units is selected we calculate its mean and put it in a set,
1
2
3
,
,
,...
x
x
x
.
By
this
process
we
obtain
a
population
of
all
sample
means:
1
2
3
,
,
,...
x
x
x
(How many elements does this population have?)
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STA3032 Fall 2011, Chapter 8, Page 2 of 12
Like any other population, the population of sample means has
A mean, called
the mean of the population of all sample means
, denoted by the
symbol
X
.
A variance called
the variance of the population of all sample means
, denoted
by
2
X
; [the square root of this variance is called the
standard error of the
sample mean, denoted by
(
)
X
SE X
]
and
A distribution called the (
sampling) distribution of sample mean
(or simply
distribution of
X
)
which is the probability distribution of the means of all
samples of size n, from the given population.
8.2 & 8.3 Distribution of
X
and
ˆ
p
Suppose a simple random sample of size n is selected from a population that has
mean μ
X
and standard deviation σ
X
. Then,
Theorem 1:
2
~
,
X
X
X
N
n
for any n if
2
~
(
,
)
X
X
X
N
, i.e., the population
from which the sample is selected has a normal distribution.
Now, suppose we know the population mean,
μ
X
, and the population standard deviation
σ
X
but
do not know the distribution of the population,
or
know that it is not normal.
Then we cannot use this theorem. Instead we will use the central limit theorem (CLT):
Theorem 2: (The Central Limit Theorem):
when the distribution population is not
known, or known to be not normal
2
~
,
X
X
X
N
n
approximately,
if n is large.
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 Summer '08
 Kyung
 Statistics, Normal Distribution, Standard Deviation, LCL

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