Random Sampling Methods and Applications

Essential Principles of Random Sampling

• All usable and valid samples must be random. Non-random samples, such as volunteers or convenience samples, are useless for statistical analysis.

• Samples must be representative of the populations from which they are drawn.

• Sampling without replacement is a strategy where an observational unit is selected only once and not returned to the population, preventing over-representation. This is the standard method in statistics.

• Sampling with replacement is a strategy where an observational unit is returned to the population and can be selected more than once.

Simple Random Sample (SRS)

• Definition: A sample in which every group of a given size has an equal chance of being selected.

• Hat Method Procedure: Place names on slips of paper of equal size, shake the hat, and select names without replacement.

• Random Number Generator Procedure:

  • Assign each individual a distinct number ($1$ to $n$).

  • Use a generator to select unique numbers.

  • Ignore repeats and numbers not assigned to individuals.

  • The people corresponding to the selected numbers form the sample.

Stratified Random Sample

• Procedure: Divide the population into separate, non-overlapping, homogeneous groups called strata based on shared characteristics (e.g., grade level or hair color).

• Select an SRS from each stratum and combine them.

• Benefit: Guarantees that specific subgroups are represented and reduces variation between the sample and the population. This is used when a variable is expected to impact survey responses.

Cluster Random Sample

• Procedure: Divide the population into heterogeneous groups called clusters, where each group ideally mirrors the diversity of the entire population.

• Randomly select one or more clusters and include all individuals within those selected clusters in the sample.

• Benefit: Faster and more efficient than other methods when populations are already naturally divided into mixed groups (e.g., school lunch periods).

Systematic Random Sample

• Procedure: Select a random starting point on a list and then select individuals at a fixed periodic interval (e.g., every $5^{th}$, $10^{th}$, or $15^{th}$ name).

• This method provides a random, representative sample without high complexity.

Application: Valley City High School Example

• Population: $1,200$ students across grades $9$ through $12$. Target sample size: $120$ students.

• SRS Method: Assign numbers $1$ to $1,200$ and select $120$ unique numbers.

• Stratified Method: Categorize by grade level and select $30$ students from each grade.

• Cluster Method: Select all students from $3$ or $4$ specific home rooms out of the $12$ available groups, assuming home rooms are mixed.

• Systematic Method: Select every $10^{th}$ student entering the cafeteria line from a random starting point.