统计代写|生物统计代写Biostatistics代考|DETERMINING THE SAMPLE SIZE

One of the most important aspects of any sampling plan is the determination of the number of units to sample. The sample size is denoted by $n$ and is determined by several factors including

• the size of the population:
• the level of reliability required in the statistical analysis;
• the cost of sampling;
• whether or not subpopulations are present in the population:
• the variability in the population.
Of all the factors affecting the sample size, the most important factor to consider is the degree of reliability that will be required of the statistical analysis. The measures of reliability for a statistical analysis will depend on the statistical methods used in the analysis; however, one of the more commonly used measures of reliability for an estimate of a population proportion or mean is the bound on the error of estimation, which is denoted by $B$. The bound on the error of estimation is based on the two standard deviation empirical rule. Moreover, it is unlikely for an estimate of a proportion or a mean to be farther than $B$ from the value of the population proportion or mean being estimated. Careful consideration must be given in prespecifying the value of $B$ in designing a sampling plan with the appropriate sample size. That is, when $n$ is too small, the estimates will not be reliable enough to provide useful information, and when the sample size is too large the benefits of having a large sample are not cost-effective.

In the following section, methods for determining the appropriate sample size for a given value of the bound on the error of estimation will be discussed only for simple random, stratified random, and systematic random samples. The determination of the sample size for a cluster sample is beyond the scope of this book but is covered in many specialized textbooks on sampling.

统计代写|生物统计代写Biostatistics代考|The Sample Size for Simple and Systematic Random Samples

In a simple random sample or a systematic random sample, the sample size required to produce a prespecified bound on the error of estimation for estimating the mean is based on the number of units in the population $(N)$, and the approximate variance of the population $\sigma^2$. Moreover, given the values of $N$ and $\sigma^2$, the sample size required for estimating a mean $\mu$ with bound on the error of estimation $B$ with a simple or systematic random sample is
$$n=\frac{N \sigma^2}{(N-1) D+\sigma^2}$$
where $D=\frac{B^2}{4}$. Note that this formula will not generally return a whole number for the sample size $n$; when the formula does not return a whole number for the sample size, the sample size should be taken to be the next largest whole number.
Example 3.11
Suppose a simple random sample is going to be taken from a population of $N=5000$ units with a variance of $\sigma^2=50$. If the bound on the error of estimation of the mean is supposed to be $B=1.5$, then the sample size required for the simple random sample selected from this population is
$$n=\frac{5000(50)}{4999\left(\frac{1.52}{4}\right)+50}=87.35$$
Since $87.35$ units cannot be sampled, the sample size that should be used is $n=88$. Also, $n=$ 88 would be the sample size required for a systematic random sample from this population when the desired bound on the error of estimation for estimating the mean is $B=1.5$. In this case, the systematic random sample would be a 1 in 56 systematic random sample since $\frac{5000}{88} \approx 56$.

In many research projects, the values of $N$ or $\sigma^2$ are often unknown. When either $N$ or $\sigma^2$ is unknown, the formula for determining the sample size to produce a bound on the error of estimation for a simple random sample can still be used as long as the approximate values of $N$ and $\sigma^2$ are available. In this case, the resulting sample size will produce a bound on the error of estimation that is close to $B$ provided the approximate values of $N$ and $\sigma^2$ are reasonably accurate.

The proportion of the units in the population that are sampled is $n / N$, which is called the sampling proportion. When a rough guess of the size of the population cannot be reasonably made, but it is clear that the sampling proportion will be less than $5 \%$, then an alternative formula for determining the sample size is needed. In this case, the sample size required for a simple random sample or a systematic random sample having bound on the error of estimation $B$ for estimating the mean is approximately
$$n=\frac{4 \sigma^2}{B^2}$$

生物统计代考

统计代写|生物统计代写Biostatistics代考|DETERMINING THE SAMPLE SIZE

• 人口规模：
• 统计分析所需的可靠性水平；
• 抽样成本；
• 人群中是否存在亚群：
• 人口的可变性。
在影响样本量的所有因素中，要考虑的最重要因素是统计分析所需的可靠性程度。统计分析的可靠性度量将取决于分析中使用的统计方法；然而，用于估计总体比例或平均值的更常用的可靠性度量之一是估计误差的界限，表示为乙. 估计误差的界限是基于两个标准差的经验法则。此外，对比例或平均值的估计不太可能超过乙从估计的人口比例或平均值的值。必须仔细考虑预先指定的值乙在设计具有适当样本量的抽样计划时。也就是说，当n太小，估计值将不够可靠，无法提供有用的信息，而当样本量太大时，拥有大样本的好处不具有成本效益。

统计代写|生物统计代写Biostatistics代考|The Sample Size for Simple and Systematic Random Samples

n=ñp2(ñ−1)D+p2

n=5000(50)4999(1.524)+50=87.35

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