3.2 Methods for selecting simple random samples

Simple random sampling is a cornerstone of survey research. It ensures every unit has an equal chance of selection, providing unbiased estimates of population parameters. This section covers various methods for implementing simple random sampling, from computerized techniques to manual approaches.

We'll explore basic and advanced random sampling techniques, including stratified and cluster sampling. We'll also discuss key considerations like sampling frames, sample size determination, and statistical concepts such as sampling error and confidence intervals. These tools are essential for conducting robust survey research.

Random Selection Methods

Computerized Random Selection Techniques

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• Random number generator produces sequences of numbers without discernible pattern
  • Utilizes complex algorithms to ensure randomness
  • Can generate integers within specified range for sample selection
  • Widely available in statistical software packages (R, SPSS, SAS)
• Computer-assisted random selection automates sampling process
  • Integrates random number generation with sampling frame
  • Allows for efficient selection of large samples
  • Reduces human error in selection process
  • Often includes features for stratification and cluster sampling

Manual Random Selection Methods

• Lottery method involves assigning unique numbers to population units
  • Numbers written on identical slips of paper or balls
  • Placed in container and thoroughly mixed
  • Drawn one by one until desired sample size reached
  • Ensures equal probability of selection for each unit
• Systematic sampling selects units at fixed intervals after random start
  • Calculate sampling interval $k = N/n$ (N = population size, n = sample size)
  • Randomly select starting point between 1 and k
  • Select every kth unit thereafter
  • Can introduce bias if population has cyclical patterns

Types of Random Sampling

Basic Random Sampling Techniques

• Simple random sampling gives equal probability of selection to all units
  • Requires complete list of population units
  • Each unit has known, non-zero probability of selection
  • Allows for unbiased estimation of population parameters
  • Can be inefficient for large, diverse populations
• Stratified random sampling divides population into homogeneous subgroups
  • Strata formed based on relevant characteristics (age, gender, income)
  • Simple random sample drawn from each stratum
  • Improves precision by reducing within-group variability
  • Allows for separate analysis of subgroups

Advanced Random Sampling Techniques

• Cluster sampling selects groups of units rather than individual units
  • Population divided into clusters (geographic areas, schools)
  • Random sample of clusters selected
  • All units within selected clusters included in sample
  • Reduces costs for geographically dispersed populations
  • May increase sampling error due to between-cluster differences
• Probability proportional to size sampling gives larger units higher selection probability
  • Selection probability proportional to measure of size (population, sales)
  • Useful when units vary greatly in size or importance
  • Improves precision for estimating population totals
  • Requires accurate size measures for all units

Sampling Considerations

Sampling Frame and Selection Process

• Sampling frame comprises list of all units in target population
  • Ideally complete, accurate, and up-to-date
  • May include duplicates, ineligibles, or missing units
  • Quality of frame affects representativeness of sample
• Replacement vs. without replacement affects selection probabilities
  • With replacement: units returned to population after selection
    • Allows for multiple selections of same unit
    • Simplifies probability calculations
  • Without replacement: units removed from population after selection
    • Ensures unique units in sample
    • Changes selection probabilities as sampling progresses

Statistical Considerations in Sampling

• Sample size determination balances precision and cost
  • Factors include desired precision, population variability, confidence level
  • Larger samples provide more precise estimates but increase costs
  • Formula: $n = \frac{z^2 \sigma^2}{E^2}$ (z = z-score, σ = population standard deviation, E = margin of error)
• Sampling error measures variability of estimates across different samples
  • Quantifies uncertainty due to sampling process
  • Decreases as sample size increases
  • Calculated as standard error: $SE = \frac{\sigma}{\sqrt{n}}$ (σ = population standard deviation, n = sample size)
• Confidence interval provides range likely to contain true population parameter
  • Typically expressed as 95% confidence interval
  • Calculated as: $CI = \hat{\theta} \pm z \times SE$ (θ̂ = sample estimate, z = z-score, SE = standard error)
• Margin of error represents maximum expected difference between sample estimate and population parameter
  • Expressed in same units as estimate
  • Calculated as: $MOE = z \times SE$ (z = z-score, SE = standard error)
  • Used to report precision of survey results (±3 percentage points)