重學 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. 來自同一批人
2. Each element is selected independently. 每一個是 random 選擇的。

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.
就是重複提取一個 simple random sample，然後計算它的平均數，重複了500次（例子）之後，得到一個distribution很類似 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，會使得sample mean 的 standard deviation越小，所以distribution會越集中。

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 後面那個公式在 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: 選則 standard error小的，因爲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.
用於城市區域的取樣本

Systematic Sampling

5000個，提取50個，先分成100組，然後在0-100選個數字，然後其他的樣本直接用加100 提取

Convenience Sampling

Convenience sampling is a nonprobability sampling technique. 用在野外生物的樣本，或者是檢測一個船上的橘子，不可能按照 probability 去取樣本

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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數學公式

使用MathJax渲染LaTex 數學公式，詳見math.stackexchange.com.

• 行內公式，數學公式爲：Γ(n)=(n−1)!∀n∈N 。
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x=−b±b2−4ac−−−−−−−√2a

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