Population, Sample, and Sampling

Stats

Population, Sample, and Sampling

데먕 2020. 1. 11. 14:59

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.