### 统计代写|生物统计代写biostatistics代考| DETERMINING THE SAMPLE SIZE

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 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
$$n=\frac{5000(50)}{4999\left(\frac{1.5^{2}}{4}\right)+50}=87.35$$
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
$$n=\frac{4 \sigma^{2}}{B^{2}}$$

## 统计代写|生物统计代写biostatistics代考|The Sample Size for a Stratified Random Sample

Recall that a stratified random sample is simply a collection of simple random samples selected from the subpopulations in the target population. In a stratified random sample, there are two sample size considerations, namely, the overall sample size $n$ and the allocation of $n$ units over the strata. When there are $k$ strata, the strata sample sizes will be denoted by $n_{1}, n_{2}, n_{3}, \ldots, n_{k}$, where the number to be sampled in strata 1 is $n_{1}$, the number to be sampled in strata 2 is $n_{2}$, and so on.

There are several different ways of determining the overall sample size and its allocation in a stratified random sample. In particular, proportional allocation and optimal allocation are two commonly used allocation plans. Throughout the discussion of these two allocation plans, it will be assumed that the target population has $k$ strata, $N$ units, and $N_{j}$ is the number of units in the $j$ th stratum.

The sample size used in a stratified random sample and the most efficient allocation of the sample will depend on several factors including the variability within each of the strata, the proportion of the target population in each of the strata, and the costs associated with sampling the units from the strata. Let $\sigma_{i}$ be the standard deviation of the $i$ th stratum, $W_{i}=N_{i} / N$ the proportion of the target population in the $i$ th stratum, $C_{0}$ the initial cost of sampling, $C_{i}$ the cost of obtaining an observation from the $i$ th stratum, and $C$ is the total cost of sampling. Then, the cost of sampling with a stratified random sample is
$$C=C_{0}+C_{1} n_{1}+C_{2} n_{2}+\cdots+C_{k} n_{k}$$
The process of determining the sample size for a stratified random sample requires that the allocation of the sample be determined first. The allocation of the sample size $n$ over the $k$ strata is based on the sampling proportions that are denoted by $w_{1}, w_{2}, \ldots w_{k}$. Once the sampling proportions and the overall sample size $n$ have been determined, the $i$ th stratum sample size is $n_{i}=n \times w_{i}$.

The simplest allocation plan for a stratified random sample is proportional allocation that takes the sampling proportions to be proportional to the strata sizes. Thus, in proportional allocation, the sampling proportion for the $i$ th stratum is equal to the proportion of the population in the ith stratum. That is, the sampling proportion for the $i$ th stratum is
$$w_{i}=\frac{N_{i}}{N}$$
The overall sample size for a stratified random sample based on proportional allocation that will have bound on error of estimation for estimating the mean equal to $B$ is
$$n=\frac{N_{1} \sigma_{1}^{2}+N_{2} \sigma_{2}^{2}+\cdots+N_{k} \sigma_{k}^{2}}{N\left[\frac{B^{2}}{4}\right]+\frac{1}{N}\left(N_{1} \sigma_{1}^{2}+N_{2} \sigma_{2}^{2}+\cdots+N_{k} \sigma_{k}^{2}\right)}$$
The sample size for the simple random sample that will be selected from the $i$ th stratum according to proportional allocation is
$$n \times w_{i}=n \times \frac{N_{i}}{N}$$

## 统计代写|生物统计代写biostatistics代考|Bar and Pie Charts

In the case of qualitative or discrete data, the graphical statistics that are most often used to summarize the data in the observed sample are the bar chart and the pie chart since the

important parameters of the distribution of a qualitative variable are population proportions. Thus, for a qualitative variable the sample proportions are the values that will be displayed in a bar chart or a pie chart.

In Chapter 2, the distribution of a qualitative variable was often presented in a bar chart in which the height of a bar represented the proportion or the percentage of the population having each quality the variable takes on. With an observed sample, bar charts can be used to represent the sample proportions or percentages for each of the qualities the variable takes on and can be used to make statistical inferences about the population distribution of the variable.

There are many types of bar charts including simple bar charts, stacked bar charts, and comparative side-by-side bar charts. An example of a simple bar chart for the weight classification for babies, which takes on the values normal and low, in the Birth Weight data set is shown in Figure 4.1.

Note that a bar chart represents the category percentages or proportions with bars of height equal to the percentage or proportion of sample observations falling in a particular category. The widths of the bars should be equal and chosen so that an appealing chart is produced. Bar charts may be drawn with either horizontal or vertical bars, and the bars in a bar chart may or may not be separated by a gap. An example of a bar chart with horizontal bars is given in Figure $4.2$ for the weight classification of babies in the Birth Weight data set.
In creating a bar chart it is important that

1. the proportions or percentages in each bar can be easily determined to make the bar chart easier to read and interpret.
2. the total percentage represented in the bar chart should be 100 since a distribution contains $100 \%$ of the population units.
3. the qualities associated with an ordinal variable are listed in the proper relative order! With a nominal variable the order of the categories is not important.
4. the bar chart has the axes of the bar chart clearly labeled so that it is clear whether the bars represent a percentage or a proportion.
5. the bar chart has either a caption or a title that clearly describes the nature of the bar chart.

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

n=ñσ2(ñ−1)D+σ2

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

n=4σ2乙2

## 统计代写|生物统计代写biostatistics代考|The Sample Size for a Stratified Random Sample

C=C0+C1n1+C2n2+⋯+Cķnķ

n=ñ1σ12+ñ2σ22+⋯+ñķσķ2ñ[乙24]+1ñ(ñ1σ12+ñ2σ22+⋯+ñķσķ2)

n×在一世=n×ñ一世ñ

## 统计代写|生物统计代写biostatistics代考|Bar and Pie Charts

1. 可以轻松确定每个条形中的比例或百分比，以使条形图更易于阅读和解释。
2. 条形图中表示的总百分比应为 100，因为分布包含100%人口单位。
3. 与序数变量相关的质量以正确的相对顺序列出！对于名义变量，类别的顺序并不重要。
4. 条形图清楚地标记了条形图的轴，以便清楚条形是代表百分比还是比例。
5. 条形图具有清楚地描述条形图性质的标题或标题。

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