12.1 Simple random sampling and stratified sampling

Sampling techniques are crucial in data collection and analysis. Simple random sampling gives everyone an equal shot at being picked, while stratified sampling divides the population into groups first. These methods help researchers gather representative data efficiently.

Both approaches have their strengths. Simple random sampling is straightforward and unbiased, while stratified sampling ensures all subgroups are represented. Choosing the right method depends on your population and research goals. Understanding these techniques is key to conducting solid studies.

Simple Random Sampling

Definition and Key Characteristics

• Simple random sampling selects members from a population giving each an equal chance of being chosen
• Process involves assigning unique identifiers to population members and using random number generators for selection
• Provides unbiased selection and population representativeness
• Allows estimation of sampling error
• Forms the foundation for statistical inference enabling population generalizations based on the sample
• Assumes population homogeneity and equal importance of each member for study objectives
• Requires a complete sampling frame
• Presents logistical challenges for large or geographically dispersed populations

Implementation and Applications

• Utilizes random number tables or computer-generated random numbers for selection
• Applies in various fields (social sciences, market research, quality control)
• Suitable for homogeneous populations where subgroup representation is not critical
• Used in lottery systems and randomized controlled trials in medical research
• Serves as a benchmark for comparing other sampling methods
• Facilitates unbiased estimation of population parameters (mean, variance, proportion)
• Allows for straightforward calculation of standard errors and confidence intervals

Advantages and Limitations

• Advantages include simplicity, ease of understanding, and minimal prior knowledge of population required
• Eliminates selection bias and ensures each combination of individuals has an equal chance of being chosen
• Supports valid statistical inference and hypothesis testing
• Limitations include potential underrepresentation of minority groups in heterogeneous populations
• May result in larger sampling errors compared to other methods, especially for small samples
• Can be impractical or expensive for large, geographically dispersed populations
• Effectiveness diminishes when the population lacks a comprehensive list or sampling frame

Stratified Sampling vs Simple Random Sampling

Stratified Sampling Concept and Process

• Divides population into mutually exclusive and exhaustive subgroups (strata) before sampling
• Ensures adequate representation of important population subgroups in the sample
• Stratification based on relevant characteristics (age, gender, income)
• Involves separate simple random sampling within each stratum
• Combines samples from all strata to form the final sample
• Requires knowledge of the stratification variable for all population members
• Can employ different sampling methods or sample sizes within each stratum

Advantages of Stratified Sampling

• Increases precision of population estimates compared to simple random sampling
• Improves representation of minority groups or subpopulations of interest
• Enables comparisons between strata (urban vs rural populations)
• Leads to smaller overall sample sizes while maintaining or improving precision
• Particularly useful for heterogeneous populations with identifiable homogeneous subgroups
• Reduces sampling error and increases efficiency of population estimates
• Allows for focused sampling in strata of particular interest or with high variability

Comparative Analysis

• Stratified sampling generally provides more precise estimates than simple random sampling
• Simple random sampling is easier to implement and requires less prior knowledge of the population
• Stratified sampling can be more cost-effective, especially when sampling costs vary between strata
• Simple random sampling may be preferable when the population is relatively homogeneous
• Stratified sampling excels in capturing the characteristics of diverse populations
• Simple random sampling is more vulnerable to chance fluctuations in sample composition
• Stratified sampling offers greater control over the sample composition and representation of subgroups

Sample Size Calculation

Factors Influencing Sample Size

• Desired precision level affects required sample size (higher precision requires larger samples)
• Confidence level impacts sample size (higher confidence levels increase sample size)
• Population variability influences sample size (more variable populations require larger samples)
• Population size affects sample size, especially for small populations
• Expected effect size in comparative studies determines required sample size
• Resource constraints (budget, time, personnel) may limit achievable sample size
• Type of study (descriptive, analytical, experimental) influences sample size considerations

Calculation Methods for Simple Random Sampling

• Utilizes formula considering standard error, margin of error, and z-score for chosen confidence level
• Sample size formula: $n = \frac{z^2 * p * (1-p)}{e^2}$ where n is sample size, z is z-score, p is population proportion, and e is margin of error
• For finite populations, applies correction factor: $n_{adjusted} = \frac{n}{1 + \frac{n-1}{N}}$ where N is population size
• Requires estimation of population proportion (p) or use of conservative 0.5 value
• Iterative process may be necessary to balance precision and feasibility
• Power analysis incorporates to ensure sufficient statistical power for detecting effects
• Specialized software (G*Power, nQuery) facilitates complex sample size calculations

Stratified Sampling Considerations

• Overall sample size typically smaller than simple random sampling for same precision level
• Accounts for number of strata and allocation method (proportional or optimal)
• Neyman allocation formula determines optimal sample sizes for each stratum
• Considers both stratum size and variability in calculations
• May require separate sample size calculations for each stratum based on specific objectives
• Balances precision requirements across strata with overall resource constraints
• Adjusts for expected response rates within each stratum to achieve desired effective sample size

Proportional vs Optimal Allocation in Stratified Sampling

Proportional Allocation

• Assigns sample sizes to strata proportional to stratum size in the population
• Formula: $n_h = n * \frac{N_h}{N}$ where n_h is stratum sample size, n is total sample size, N_h is stratum population size, and N is total population size
• Simpler to implement than optimal allocation
• Used when limited information about stratum variability is available
• Maintains population proportions in the sample, simplifying weighting in analysis
• Suitable when strata have similar variability or when simplicity is prioritized
• May be less efficient for strata with high variability but small population sizes

Optimal (Neyman) Allocation

• Assigns sample sizes based on both stratum size and variability
• Formula: $n_h = n * \frac{N_h * S_h}{\sum N_i * S_i}$ where S_h is the standard deviation within stratum h
• Generally leads to more precise estimates than proportional allocation
• Particularly effective when there are significant differences in variability between strata
• Requires knowledge or estimation of within-stratum standard deviations
• May result in very small or large samples for certain strata, potentially causing implementation issues
• Optimizes resource allocation by focusing larger samples on more variable strata

Comparative Analysis and Decision Factors

• Optimal allocation typically provides greater precision for the same overall sample size
• Proportional allocation maintains representation of population subgroups in the sample
• Choice depends on study objectives, available resources, and knowledge of population characteristics
• Optimal allocation may be preferred in studies focusing on precise overall population estimates
• Proportional allocation might be chosen for studies emphasizing subgroup comparisons
• Hybrid approaches combining elements of both methods can be used in some situations
• Consider practical constraints (cost per sample in each stratum, accessibility) when choosing allocation method