利用总体变异系数特征预先确定调查所需样本量的方法

∑ we usually use the sample mean (
x
=
1 n
n i =1
xi
) to infer the population mean(notated as X
), and use the
variance of x to evaluate the effect of that inference, which appears as Var(x ) = N − n S 2 (here, N and
exists
d
=
rX
.Similarly
,we
can
let
V＝⎛⎜ ⎝
rX uα
⎞2 ⎟ and ⎠
get
n0
=
⎜⎛ uα S ⎝ rX
⎟⎞2 .As ⎠
S X
is the coefficient
of variation (notated as CV )of the population, we can get the following equation,
Therefore, to all the t ≥ 0 , the value range of CV of the left-skewed right triangle distribution is (0,0.3535].
f(x)
f(x)
0t
EX t+ R x 0
t EX
t+ R x
Figure 1 The Left-skewed and the Right-skewed Right Triangle Distributions 2.1.2 The right-skewed right triangle distribution. Similarly, to the right-skewed right triangle distribution (see the right one of Figure 1 ),it exists,
nN
S 2 denote the size and variance of the population respectively, and n denotes the sample size ).The
determination of the appropriate sample size depends on the specific type of accuracy requirements and the
nN
nN
sample
size
needed
and
get
n = S2 /V 1+ S 2 / NV
.
Let
n0
=
S2 V
,then
n = n0 1 + n0 / N
.And
when N
is
large enough, it exists n ≈ n0 ;
b) As the accuracy requirement is expressed as the up limit of absolute error (notated as d ) of x ,that is
1 Introduction
At the phase of sampling design in sample surveys, it is often required to decide the appropriate sample size
before the execution of the survey. Take the simple random sampling without replacement for instance ,
Abstract In sampling design ,the necessary sample size often needs to be determined before the execution of the survey. Using the up limit of coefficient of variation (CV)of the sampled population to compute the needed approximate sample size is one of the most effective and convenient methods. In fact, to common populations , it does exist some up limits for their CVs. So, this paper aims to find these up limits of CV for them to facilitate the determination of the appropriate sample size in advance. Keywords: Coefficient of variation ;Sample size; Sample surveys; Statistics
variation of the target population,
a) As the accuracy requirement of that inference is expressed as the up limit of variance of x ,which is
notated asV ,namely, Var(x ) = N − n S 2 ≤ V , we can use N − n S 2＝V to compute the maximum
to
say
Var(x )
≤
⎛ ⎜ ⎝
d uα
⎞2 ⎟. ⎠
According
to
the
conclusions
in
case
a),we
can
let
V＝⎛⎜ ⎝
d uα
⎞2 ⎟ ⎠
and
get
1
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n0
=
⎜⎜⎝⎛
uα S d
⎟⎟⎠⎞2 .
c) When the accuracy requirement is expressed as the up limit of relative error (notated as r )of x , it
R ,denotes the range of population under consideration;
CVmax ,denotes the up limit of coefficient of variation of the population under consideration.
2
In our daily lives ,most of the real populations are non-negative, such as Height or Weight of the Chinese Adult men , Total Annual Revenue of Households in certain region ,and so on. The minimum values of these populations are usually positive or equal to zero at least . Therefore, this paper assumes that the
164 West Xingang Road , Guangzhou, P.R. China, 510301 yefengwu@126.com
School of Statistics of Renmin University of China, Beijing, P.R. China,100872 duzifang@sohu.com
n0
=
⎜⎛ uα ⎝r
⎟⎞2 (CV )2 ⎠
(1)
Whether the accuracy requirement is expressed as V , d or r ,it is given by designers in advance. In order to get the appropriate sample size n , the variation(appears as population’s variance , notated as S 2 orσ 2 ;or population’s coefficient of variation , abbreviated as CV ) of the target population needs to
minimum values （notated as t ） of all the debated distributions are not less than zero, namely, t ≥ 0 .
The following sections will debate the features ,specifically, the up limits of CVs of some special distributions at first, then shift to those of the more common and more complicated distributions. We can use these up limits as conservative estimates of CVs of the target populations .
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2 Features of CVs of various special distributions
2.1 The right triangle distribution
2.1.1 The left-skewed right triangle distribution
Besides the notations mentioned above, we introduce another set of notations to facilitate the illustration,
ห้องสมุดไป่ตู้
X ,denotes the population under consideration; EX ,denotes the expectation or mean of the population under consideration;
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Sample size determined method using features of CV of the sampled population
Yefeng Wu*
Zifang Du
South China Sea Institute of Oceanology of Chinese Academy of Sciences ,
be known .However, it is usually unknown before the execution of the survey, and needs to be estimated approximately through relevant information by the designer. That is so-called ‘estimating the variance (or variation) of the sampled population in advance’. Here ,we focus on case c) to find the approaches of estimating CVs of common populations to use equation (1) to compute the needed sample size rapidly in advance.
See the left one of Figure 1 ,the real line segment is the probability density curve of the left-skewed
distribution with a range of R . According to the nature of probability density curve ,we can get its
expectation EX = t + 2R = t + 0.6667R ,and its variance σ 2 = 0.056R2 ⇒ σ = 0.2357R .So, its 3
σ
CV is
=
0.2357R
≤ 0.3535 .when t = 0 , CV = 0.3535 .
EX t + 0.6667R