Sampling Formulae
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Sampling formulae Simple Random Sampling Estimate of Population Mean µ 1 ˆ n i i y y n µ = = = ∑ (4.1) Estimate of variance of mean y 2 ˆ () s N n V y n N − = ÷ (4.2) Where 2 2 1 ( ) 1 n i i y y s n = − = − ∑ Boun on the error of estimation ! 2 ˆ 2 () 2 s N n V y n N − = ÷ (4.") Estimator of the population total τ (SRS) 1 ˆ n i i N y Ny n τ = = = ∑ (4.4) Estimate variance of τ 2 2 ˆ ˆ ˆ () ( ) s N n V V Ny N n N τ − = = ÷ ÷ (4.#) Boun on the error of estimation 2 2 ˆ 2 ( ) 2 s N n V Ny N n N − = ÷ ÷ (4.$) Sample si%e re&uire to estimate µ 'ith a oun on the error of estimation B 2 2 ( 1) N n N D σ σ = − + (4.11) 'here *! 2 4 B 1
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Sampling Formulae
Transcript of Sampling Formulae
ESEE3101/ES327 Rekabentuk Pensampelan dan Analisis Data -- Kuliah 7
Sampling formulaeSimple Random SamplingEstimate of Population Mean (
(4.1)Estimate of variance of mean
(4.2)
Where
Bound on the error of estimation = (4.3)Estimator of the population total ( (SRS)
(4.4)
Estimated variance of (
(4.5)Bound on the error of estimation
(4.6)
Sample size required to estimate with a bound on the error of estimation B:
(4.11)
where D=
Sample size required to estimate with a bound on the error of estimation B:
(4.13)
where D=
Estimation of Population Proportion based on SRSyi = 1if the element possess a certain characteristic, 0 if not.
(4.14)
Estimated variance of :
(4.15)
Bound on the error of estimation
(4.16)
Sample size required to estimate p with a bound on the error of estimation B:
(4.18) where q=1-p dan D =
Stratified Random Sampling
Estimator of the population mean
(5.1)
Estimated variance of
(5.2)
Estimator of the population total (
(5.3)
Estimated variance of the population total ( =
= (5.4)
=
ni = n x ai where ai is the fraction of observations allocated to stratum i. (5.5)Approximate sample size required to estimate or with a bound B on the error of estimation.
(5.6)where ai = ni/n
and D= B2 /4 for estimating mean (
D=B2 /4N2 for estimating total (Approximate allocation that minimizes cost for a fixed value of V () or minimum
V () for a fixed cost:
(5.7)
(5.8)
Neyman allocation
(5.9)
Under Neyman allocation,
(5.10)
Proportional allocation
(5.11)
Under proportional allocation:
(5.12)
Estimator of the population proportion p:
(5.13)
Estimated variance of
(5.14)Approximate sample size required to estimate p with a bound B on the error of estimation:
(5.15)
Approximate allocation that minimizes cost for a fixed value of or minimizes for a fixed cost:
(5.16)
Systematic Sampling
Estimator of the population mean
(7.1)
Estimated variance of
(7.2)
Estimator of the population total
EMBED Equation.3 (7.5)
Estimated variance of
(7.6)
Estimator of the population proportion p
(7.7)
Estimated variance of
(7.8)
Sample size required to estimate p with a bound B on the error of estimation:
(7.10)where
Sample size required to estimate p with a bound B on the error of estimation:
(7.11)where q=1-p and
Cluster Sampling
Estimator of the population mean
(8.1)
Estimated variance of
(8.2)
where
(8.3)
Estimator of the population total
(8.4)
Estimated variance of
(8.5)If M is not known, then
(8.6)
Estimator of the population , which does not depend on M:
(8.7)
Estimated variance of N
(8.8)
where
(8.9)
Approximate sample size required to estimate , with a bound B on the error of estimation:
(8.12)
where is estimated by and D=
Approximate size required to estimate , using M, with a bound B on the error of estimation.
(8.13)
where is estimated by , and
Approximate sample size required to estimate , using , with a bound B on the error of estimation:
(8.15)
where is estimated by and
Estimator of the population proportion p:
(8.16)
Estimated variance of
(8.17)
where
(8.18)
Cluster sampling with probabilities proportional to size
Estimator of the population mean :
(8.19)Estimated variance of
(8.20)
Estimator of the population total :
(8.21)Estimated variance of
(8 .22)
Two-Stage Cluster Sampling
Unbiased estimator of the population mean
(9.1)
Estimated variance of
(9.2)
(9.3)
i=1, 2,3,n (9.4)
Estimation of the population total (
( 9.5)
Estimated variance of
(9.6)
Ratio estimator of the population mean
(9.7)
Estimated variance of
(9.8)
where
(9.9)
and
n=1,2,.....n (9.10)
Estimator of a population proportion p:
(9.11)
Estimated variance of p:
(9.12)
where
(9.13)
Sampling equal-size clusters
(9.14)
9.2 now becomes
(9.15)
where f1= n/N and
When N is large
(9.16)
(9.17)
where = variance among the cluster means
= variance among the elements within clusters
(9.19)
where c1 is associated with cost of sampling each cluster and c2 is associated with cost of sampling each element within a cluster, and c is the total cost.The value of m that minimizes for fixed cost, or minimizes c for fixed variance is
(9.20)
Two-Stage Cluster Sampling with Probabilities Proportional to Size
Estimator of population mean
(9.22)
Estimated variance of
(9.23)Estimator of the population total
(9.24)Estimated variance of
(9.25)
Control of Sub-sample size
f =
b= number of cases that we want to select, and Ni is the cluster size for the ith cluster, f is the sampling fraction and F is the inverse of f.
[N( / F x b] x [b x N(] = 1/F = f
First stage second stage overall fraction sampling fraction
Design effects
_ _
Var (yst) / Var (y srs) for stratified random sample, and its value is less than 1 if the design is more efficient than the srs
_ _
Var (ycl) / Var (ysrs) for cluster samplingIf one has the estimate of the design effect, and the variance from srs, one can estimate the variance from complex sample, simply by
_ _
Var (ycl) = deft2 [Var (ysrs)]
_
Deft2 = 1+ ( (n-1)
_
( = [deft2 -1]/ [n-1]
where ( =intraclass homogeneity
_
n = average number of elements in the selected cluster
This implies that if the sample size if srs is reduced by half, the varians will be double, given
_
Var (y) = s2 / n
_ _
Thus, if s2 is constant, Var (y) =constant/100 will be that of Var (y) = constant/200.
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