Description and Properties of simple random sampling

 Description of Simple Random Sampling

Simple random sampling (SRS) is a probability sampling method in which every member of the population has an equal chance of being selected for the sample. This is the most basic type of probability sampling and is often used as a benchmark for other sampling methods.

Properties of Simple Random Sampling

SRS has a number of desirable properties, including:

• Unbiasedness: SRS samples are unbiased, meaning that they have the same expected value as the population they are drawn from.
• Efficiency: SRS is a relatively efficient sampling method, meaning that it requires a smaller sample size than other sampling methods to achieve the same level of precision.
• Simplicity: SRS is a simple sampling method to implement, both in theory and in practice.

Mathematical Proofs of the Properties of SRS

Unbiasedness:

To prove that SRS samples are unbiased, we can use the following mathematical argument:

Let X be the population mean and S be the sample mean. Then, the expected value of the sample mean is given by:

E(S) = n/N * Σ(x_i)

where n is the sample size, N is the population size, and x_i is the value of the variable of interest for the ith individual in the population.

Under SRS, each member of the population has an equal chance of being selected for the sample. Therefore, the expected value of each x_i in the sample is equal to the population mean X. This means that the expected value of the sample mean S is also equal to the population mean X.

Efficiency:

To prove that SRS is a relatively efficient sampling method, we can compare the variance of the sample mean under SRS to the variance of the sample mean under other sampling methods.

The variance of the sample mean under SRS is given by:

Var(S) = σ^2 / n

where σ^2 is the population variance.

The variance of the sample mean under other sampling methods, such as stratified sampling, is typically smaller than the variance of the sample mean under SRS. However, SRS is still a relatively efficient sampling method, especially when the population is large.

Simplicity:

SRS is a simple sampling method to implement, both in theory and in practice.

To select a SRS sample, we can use the following steps:

1. Number each member of the population from 1 to N.
2. Generate n random numbers between 1 and N.
3. Select the members of the population whose numbers were chosen in step 2 for the sample.

Examples of Simple Random Sampling

Here are some examples of SRS:

• A researcher wants to estimate the average height of all students at a university. The researcher could use SRS to select a sample of students from the university directory and then measure their heights.
• A marketing company wants to know what percentage of consumers are interested in a new product. The company could use SRS to select a sample of consumers and then survey them about their interest in the new product.
• A government agency wants to estimate the unemployment rate in the country. The agency could use SRS to select a sample of people and then ask them about their employment status.

Conclusion

Simple random sampling is a versatile and powerful sampling method. SRS is unbiased, efficient, and simple to implement. It is often used as a benchmark for other sampling methods.