Sampling Methods and Simple Random Sampling in Statistics for Business

Sampling Methods

Simple Random Sampling: Concepts and Applications

Simple random sampling is a fundamental technique in statistics for selecting a subset of individuals from a population, ensuring that every possible sample has an equal chance of being chosen. This method is crucial for unbiased data collection and valid statistical inference.

• Definition: Simple random sampling is a sampling method where each possible sample of a given size has an equal probability of being selected from the population.

• Application: Used in survey research, quality control, and experimental design to ensure representativeness.

Enumerating Simple Random Samples

Given a finite population, it is possible to list all possible simple random samples of a specified size. This enumeration helps in understanding the probability structure of sampling.

• Example: For a population with five elements labeled A, B, C, D, and E, the possible samples of size 2 are:

Sample Number	Sample Elements
1	AB
2	AC
3	AD
4	AE
5	BC
6	BD
7	BE
8	CD
9	CE
10	DE

• Formula for Number of Samples: The number of possible samples of size $n$ from a population of size $N$ is given by: $\binom{N}{n}$ For $N=5$, $n=2$: $\binom{5}{2} = 10$

Probability of Selecting a Sample

In simple random sampling, each sample has an equal probability of being selected.

• Probability Formula: For $k$ possible samples, the probability of selecting any one sample is: $P = \frac{1}{k}$ For 10 samples: $P = \frac{1}{10} = 0.1$

• Example: Each sample of size 2 from the five-element population has a probability of 0.1 of being selected.

Using Random Numbers for Sampling

Random numbers are often used to select samples in practice, ensuring unbiased selection.

• Procedure: Assign each population element a unique number. Use random digits to select elements.

• Example: If random number 1 corresponds to A, 2 to B, etc., and the random digits are 8 0 5 7 5 3 2, select the elements corresponding to the first two valid digits.

• Application: This method is used in computer-generated sampling and lottery-style selection.

Sampling from Larger Populations

For larger populations, random numbers are used to select samples efficiently.

• Example: For a population of 350 elements, use the last three digits of five-digit random numbers to select sample elements.

Random Number	Last Three Digits	Selected Element
98601	601	Element 601
73022	022	Element 22
83448	448	Element 448
02147	147	Element 147
34229	229	Element 229
27553	553	Element 553 (Additional info: If population size is 350, ignore numbers above 350)
84147	147	Element 147 (Duplicate, select next unique)
93289	289	Element 289
14209	209	Element 209

• Note: If a selected number exceeds the population size or is a duplicate, continue to the next random number.

Summary Table: Steps in Simple Random Sampling

Step	Description
1	Assign numbers to each population element
2	Generate random numbers
3	Select elements corresponding to random numbers
4	Ensure no duplicates and numbers within population size

• Example Application: Market research, employee surveys, product quality testing.

Additional info: In practice, computer software is often used to automate random sampling, especially for large populations.