重学 Statistics，Cha7 Sampling and Sampling Distribution

7.1 Selecting a Sample

1. Simple Random Sampling (Finite)

A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.
- Sampling without replacement: any previously used random numbers are ignored because the corresponding man- ager is already included in the sample
- Sampling with replacement: selected a sample such that previously used random numbers are acceptable and specific managers could be included in the sample two or more times

2. Random Sampling (Infinite)

A random sample of size n from an infinite population is a sample selected such that the following conditions are satisfied.
1. Each element selected comes from the same population. 来自同一批人
2. Each element is selected independently. 每一个是 random 选择的。

7.3 Point Estimation

7.4 Introduction to Sampling Distribution

The various possible values of are the result of different simple random samples, the probability distribution of is called the sampling distribution of x̅.
就是重复提取一个 simple random sample，然后计算它的平均数，重复了500次（例子）之后，得到一个distribution很类似 normal distribution

Sampling Distribution of x̅
The sampling distribution of x̅ is the probability distribution of all possible values of the sample mean x̅.

Expected Value of x̅

Standard Deviation of x̅

Form of the Sampling Distribution of x̅

Population- normal distribution
When the population has a normal distribution, the sampling distribution of is normally distributed for any sample size.
Population- not normal distribution
Central Limit Theorem:
In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large.

Relationship Between the Sample Size and the Sampling Distribution

Sample Size越大，会使得sample mean 的 standard deviation越小，所以distribution会越集中。

7.6 Sample Distribution of p

The sample proportion p¯ is the point estimation of population propotion p. The formula is p¯ = x/n
The sampling distribution of p¯ is the probability distribution of all possible values of the sample proportion p¯.

E( p¯) = p

With n/N < 0.05, will use 后面那个公式在 infinite population

The sampling distribution of can be approximated by a normal distribution whenever
np>=5 and n(1 - p) >=5.

7.7 Properties of Point Estimators

How to use sample statistics to estimate the population?
θ = the population parameter of interest
θˆ = the sample statistic or point estimator of θ

Unbiased: E(θˆ) = θ

Efficiency: 选则 standard error小的，因为it tends to provide estimates closer to the population parameter.
不太理解？

Consistency: sample size要大一些

7.8 Other Sampling Methods

Stratified Random Sampling

In stratified random sampling, the elements in the population are first divided into groups called strata, such that each element in the population belongs to one and only one stratum.

Cluster Sampling

In cluster sampling, the elements in the population are first divided into separate groups called clusters. Each element of the population belongs to one and only one cluster.
A simple random sample of the clusters is then taken. All elements within each sampled cluster form the sample.
用于城市区域的取样本

Systematic Sampling

5000个，提取50个，先分成100组，然后在0-100选个数字，然后其他的样本直接用加100 提取

Convenience Sampling

Convenience sampling is a nonprobability sampling technique. 用在野外生物的样本，或者是检测一个船上的橘子，不可能按照 probability 去取样本

Judgment Sampling

the person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population.
For example, a reporter may sample two or three senators, judging that those senators reflect the general opinion of all senators. However, the quality of the sample results depends on the judgment of the person selecting the sample.

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脚注

生成一个脚注1.

目录

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数学公式

使用MathJax渲染LaTex 数学公式，详见math.stackexchange.com.

• 行内公式，数学公式为： Γ(n)=(n−1)!∀n∈N 。
• 块级公式：

x=−b±b2−4ac−−−−−−−√2a

更多LaTex语法请参考 这儿.

UML 图:

可以渲染序列图：

Created with Raphaël 2.1.0 张三 张三 李四 李四 嘿，小四儿, 写博客了没? 李四愣了一下，说： 忙得吐血，哪有时间写。

或者流程图：

Created with Raphaël 2.1.0 开始 我的操作 确认？ 结束 yes no

• 关于 序列图 语法，参考 这儿,
• 关于 流程图 语法，参考 这儿.

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1. 这里是 脚注 的 内容. ↩