# 统计代写|抽样调查作业代写sampling theory of survey代考|STAT392

## 统计代写|抽样调查作业代写sampling theory of survey代考|DETERMINATION OF SAMPLE SIZE

In any large- or small-scale survey, one has to determine the sample size, which will fulfill the objective of the survey. To get a representative sample, it is expected that the sample size should increase with the population size. In general, the sampling error decreases with the increase of sample size. However, the cost of a survey increases with the increase of sample size. The cost also increases with the increase of the size of the questionnaire because the investigator requires more time to complete a long questionnaire. The size of the sample is determined, considering available cost, time, and the degree of precision needed for estimation of parameters under consideration. In this section we will consider how one can determine an appropriate sample size for SRSWR and SRSWOR sampling designs.

Suppose the cost of a survey is expressed by a simple cost function $C=C_o+c n$ where $C_o$ is the overhead cost of the survey, which is fixed for a survey, i.e., independent of the sample size, and $c$ is the average cost for surveying a unit. Hence for a fixed available $\operatorname{cost} C=C^$ of a survey, one should select sample size $n=\left(C^-C_o\right) / c$.

The magnitude of the sampling error of an estimator is determined by its variance. Hence, one can determine the sample size that yields some specific value of variance, $V_0$, for example. To estimate the population mean $\bar{Y}$, if one uses the sample mean as an estimator and expects that the sample mean will have a specific value of its variance $V_0$, then $n$ can be derived from the relation
$$V{\bar{y}(s)}=\left(\frac{1}{n}-\frac{1}{N}\right) S_y^2=V_0 \text { i.e., } n=\left(\frac{V_0}{S_y^2}+\frac{1}{N}\right)^{-1}(\text { for SRSWOR })$$ and
$$V\left{\bar{\gamma}\left(s_o\right)\right}=\frac{1}{n} \sigma_\gamma^2=V_0 \text { i.e., } n=\frac{\sigma_\gamma^2}{V_0} \quad \text { (for SRSWR) }$$
To determine the value of $n$, one needs to know the value of $S_y^2$ ( or $\sigma_y^2=\frac{N-1}{N} S_y^2$ ), which is generally unknown. Hence, it is customary to use a value of $S_\gamma^2$ either from the past survey (or experience) or replace $S_\gamma^2$ by its estimate $s_y^2$ determined from a pilot survey.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Given Margin of Permissible Error

Here, sample size is determined by assigning the probability to a certain level, $(1-\alpha)$ (say) for the maximum permissible error (difference between the estimated and the true value of the parameter) to a certain value $d$. For instance, let $d$ be the permissible error, maximum acceptable difference between the estimator $t$, and the population mean $\bar{Y}$. The sample size is determined from the relation
$$\operatorname{Prob}{|t-\bar{Y}| \leq d}=1-\alpha$$
The above equation is equivalent to
$$\text { Prob }\left{\frac{|t-\bar{Y}|}{\sqrt{V(t)}} \leq \frac{d}{\sqrt{V(t)}}\right}=1-\alpha$$
Assuming the sample size to be so large, enabling $z=\frac{|t-\bar{Y}|}{\sqrt{V(t)}}$ distributed $N(0,1)$, one can determine the value of $n$ using the relation
$$\frac{d}{\sqrt{V(t)}}=z_{\alpha / 2}$$
For an SRSWOR design with $t=\bar{y}(s)$, Eq. (3.5.1) yields
$$n=\left[\frac{1}{N}+\left(\frac{d}{S_Y z_{\alpha / 2}}\right)^2\right]^{-1}=\left[\frac{1}{N}+\frac{N-1}{N}\left(\frac{k}{C_\gamma z_{\alpha / 2}}\right)^2\right]^{-1}$$
where $k=d / \bar{Y}$
For an SRSWR design with $t=\gamma\left(s_e\right)$, Eq. (3.5.1) yields
$$n=\left(\frac{\sigma_\gamma z_{\alpha / 2}}{d}\right)^2=\left(\frac{C_\gamma z_{\alpha / 2}}{k}\right)^2$$

# 抽样调查代考

## 统计代写|抽样调查作业代写sampling theory of survey代考|DETERMINATION OF SAMPLE SIZE

$$V \bar{y}(s)=\left(\frac{1}{n}-\frac{1}{N}\right) S_y^2=V_0 \text { i.e., } n=\left(\frac{V_0}{S_y^2}+\frac{1}{N}\right)^{-1}(\text { for SRSWOR })$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Given Margin of Permissible Error

$$\operatorname{Prob}|t-\bar{Y}| \leq d=1-\alpha$$

$$\frac{d}{\sqrt{V(t)}}=z_{\alpha / 2}$$

$$n=\left[\frac{1}{N}+\left(\frac{d}{S_Y z_{\alpha / 2}}\right)^2\right]^{-1}=\left[\frac{1}{N}+\frac{N-1}{N}\left(\frac{k}{C_\gamma z_{\alpha / 2}}\right)^2\right]^{-1}$$

$$n=\left(\frac{\sigma_\gamma z_{\alpha / 2}}{d}\right)^2=\left(\frac{C_\gamma z_{\alpha / 2}}{k}\right)^2$$

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