## 统计代写|概率与统计作业代写Probability and Statistics代考|MATH352

2022年9月29日

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• Statistical Inference 统计推断
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## 统计代写|概率与统计作业代写Probability and Statistics代考|Stratified Sampling

Simple random and systematic sampling methods implicitly assume that there is no particular group structure present in the population. At best they assume that the order of participants is aligned with an ordering in time. Structures in a population may be caused by particular characteristics. For instance, products may be manufactured on several different production lines, which form particular subpopulations within the population of products. Or, age, gender, and geographic area may form typical subgroups of people with different disease prevalence or incidence. ${ }^{20}$

When the numbers of units across these subpopulations are (substantially) different, simple random and systematic sampling may not collect units from each subgroup. Indeed, suppose that two production lines, say A and B, are used for the manufacturing of products and production line A produced 1,000 products, while production line $\mathrm{B}$ only produced 10 products. Then a simple random sample of 100 products may not necessarily contain any units from production line $\mathrm{B}^{21}$ Thus production line B would probably be under-represented. Stratified sampling is used to accommodate this issue by setting the sample size for each subpopulation (often called strata) to a fixed percentage of the number of units of the subpopulation. For instance, a $10 \%$ sample from the population of products from the two production lines $\mathrm{A}$ and $\mathrm{B}$ results in a simple random sample of size 100 units from line $\mathrm{A}$ and a simple random sample of size 1 unit from line $\mathrm{B}$. By selecting $10 \%$ of the units from each stratum we are certain that each stratum is included in the sample.

Stratified sampling can also be applied to time periods, similar to systematic sampling. The population is then divided into $n$ groups such that the order in units is maintained. From each group one unit is randomly collected with probability $1 / \mathrm{m}$, when each group contains $m$ units. Note that this form of stratified sampling is not identical to systematic sampling (although this method of stratified sampling is sometimes referred to as systematic sampling). In the case of the ordered population of six units in the example for systematic sampling, the population is again split up into three periods, i.e., $(1,2) ;(3,4)$; and $(5,6)$. With stratified sampling the sample can now exist of the following possibilities: $S_6=(1,3,5), S_7=(1,3,6), S_8=(1,4,5)$, $S_9=(1,4,6), S_{12}=(2,3,5), S_{13}=(2,3,6), S_{14}=(2,4,5), S_{15}=(2,4,6)$, using the notation of the sampling sets for simple random sampling. Thus stratified sampling may lead to samples that are not possible with systematic sampling, but it does not produce all possible samples from simple random sampling.

## 统计代写|概率与统计作业代写Probability and Statistics代考|Cluster Sampling

Directly sampling units from populations is not always feasible. For instance, in several countries there are no complete or up-to-date lists of all houses in a certain geographic area. However, using maps of the region, groups or clusters of houses can be identified and these clusters can then be sampled. In other settings, economic considerations are used to form clusters of units that are being sampled. To determine how many hours per day children in the Netherlands play video games, it is logistically easier and financially cheaper to sample schools from the Netherlands and then contact (a random sample of) the children at these schools. Thus cluster sampling involves random sampling of groups or clusters of units in the population. Cluster sampling can be less representative than sampling units directly. For instance, a random sample of 20,000 children from the Netherlands may cover the Netherlands more evenly than a random sample of 20 schools with on average 1,000 students. Additionally, cluster sampling introduces a specific structure in the sample which should also be addressed when the data is being analyzed. The cluster structure introduces two sources of variation in the data being collected. In the example of the number of hours per day that children play video games, children within one school may be more alike in their video game behavior than children from other schools.

These sources of variation need to be quantified to make proper statements on the population of interest.

Cluster sampling can be performed as single-stage or in multiple stages. A singlestage cluster sample uses a random sample of the clusters and then all units from these clusters are selected. In a two-stage cluster sample, the units from the sampled clusters are also randomly sampled instead of taking all units from the cluster. The number of stages can increase in general to any level, depending on the application. For instance, sampling children from the Netherlands can be done by sampling first a set of counties, then a set of cities within counties, then a set of schools within cities, and then finally a set of classes within schools (with or without sampling children from these classes). The sampling units for the first stage (e.g. counties) are referred to as primary cluster units. Sampling these different levels of clusters can be performed using simple random sampling, systematic sampling, or even stratified sampling, if certain cluster are put together on certain criteria.

Cluster sampling is in a way related to stratified sampling. For instance, in a twostage cluster sample, the clusters may be viewed as strata, but instead of collecting units from each stratum, the strata themselves are first being randomly sampled. Since we deal with multiple levels of hierarchical clusters, the calculation of the probability of collecting one unit from the population and the probability of collecting one of the many sample sets is more cumbersome for cluster sampling. Therefore, we do not provide general formulae.

# 概率与统计作业代考

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## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。