Population, Sample, and Sampling

1. Overview

How to select good random samples from the population

How to calculate the estimate of a population mean using samples

How can I assure that the sample will provide adequate information about the population parameters

2. Description

2.1 Simple Random Sampling Method

2.1.1 finite population

A simple random sample of size n from a finite population of size N is a sample selected such that:

• Every observation has an equal chance of being selected
• Every sample of n observation has an equal chance of being selected

2.1.2 infinite population

Select a random sample such that each element is selected independently: Avoid selection bias

2.2 Point estimators

To estimate population parameters we compute sample parameters

2.2.1 Standard Deviation

If the sample size is less than or equal to 5% of the population size $\frac{n}{N}\leq 0.05$:

$$\sigma _{\bar{x}}=\frac{\sigma }{\sqrt{n}}$$

If the sample size is greater than 5% of the population size $\frac{n}{N}$>0.05:

$$\sigma _{\bar{x}}=\frac{\sqrt{N-n}}{\sqrt{N-1}}\frac{\sigma }{\sqrt{n}}$$

where $\frac{\sqrt{N-n}}{\sqrt{N-1}}$ is finite population correction factor

2.3 Sampling Error

The standard error is a measure of the sampling error of the sampling distribution. There are others, but standard error is, by far, the most commonly used when dealing with survey data.

$$STDERR=\frac{\sigma}{\sqrt{n}}$$

$$STDERR=\sqrt{\frac{p*(1-p)}{n}}$$

you can see other formulas within this link

2.4 Central Limit Theorem

The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of $\bar{x}$ will resemble a normal distribution.

2.5 Degree of freedom

3. Reference

https://www.youtube.com/watch?v=92s7IVS6A34

https://statisticsbyjim.com/hypothesis-testing/degrees-freedom-statistics/

https://www.investopedia.com/terms/t/tdistribution.asp

https://corporatefinanceinstitute.com/resources/knowledge/other/kurtosis/

https://blog.minitab.com/blog/statistics-and-quality-data-analysis/what-are-degrees-of-freedom-in-statistics

https://m.blog.naver.com/PostView.nhn?blogId=s2ak74&logNo=220616766539&proxyReferer=https%3A%2F%2Fwww.google.com%2F

https://en.wikipedia.org/wiki/Central_limit_theorem

https://www.en-net.org/question/768.aspx

https://www.youtube.com/watch?v=uGuWrPFStdg

https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/what-is-the-standard-error-of-a-sample/