A. Zhensykbaev
Sampling distributions
Sampling distributions.
Sampling theory is a study of relationships existing between a population
and samples drawn from the population. It is of great value in many connec
tions. It is useful, for example, in estimating unknown population quantities
(such as population mean and variance), often called population parameters
or briefly parameters, from a knowledge of corresponding sample quantities
(the sample mean
X
, sample variance
s
2
, sample standard deviation
s
),
often called sample statistics or briefly statistics.
Parameter
is a numerical measure that describes a characteristic of the
population.
Sample Statistic
is a numerical measure computed from a sample to
describe a characteristic of the population.
Population parameters are usually unknown. We use sample statistics to
estimate the parameters of a population.
Sampling theory is also useful in determination whether the observed
differences between two samples are due to chance variation or whether
they are really significant. The answers involve the use of socalled test of
significance and hypotheses that are important in the theory of decisions.
In general, a study of the inferences made concerning a population by
using samples drawn from it, together with indications of the accuracy of
such inferences by using probability theory, is called statistical inference.
Sample statistics are themselves random variables. The probability distri
bution of a sample statistic (the mean, the standard deviation) is called the
sampling distribution
for the statistic.
Example
of a sampling distribution. The population consists of the
measurements
{
0
,
2
,
10
}
with equal probabilities:
P
(0) =
P
(2) =
P
(10) =
1
3
.
A random sample of 2 measurements is selected from the population. Find
sampling distribution of the sample mean
x
.
Solution
. 1. List possible samples and its mean (it is presented in the
table below). For example if
x
= (0; 2)
x
=
0 + 2
2
= 1
1
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A. Zhensykbaev
Sampling distributions
Sample
0; 0
0; 2
0; 10
2; 0
2; 2
2; 10
10; 0
10; 2
10; 10
Mean
x
0
1
5
1
2
6
5
6
10
2. Determine probability each sample: sample space consists of 9 samples.
Hence the probability of each of them equals to 1/9.
3. Determine the number of simple events containing in the event: becau
se the mean
x
= 0
occurs in one sample,
P
(
x
= 0) = 1
/
9
; similarly
x
= 1
occurs in two samples and
P
(
x
= 1) = 2
/
9
etc. So, we have the sampling
distribution of the sample mean
x
Mean
x
0
1
2
5
6
10
P
(
x
)
1/9
2/9
1/9
2/9
2/9
1/9
If a sample statistic has a sampling distribution with the mean equal to
the population parameter the statistic is intended to estimate, the statistic
is called an
unbiased estimator
of the parameter.
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 Spring '13
 ChristopherStocker
 Math, Normal Distribution, Standard Deviation, Variance, Probability theory, A. Zhensykbaev

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