Sampling Techniques and Sampling Distributions
Sampling Techniques and Sampling Distributions tests how a sample is drawn, how precise the sample mean is under different designs, and the standard distributions used to describe sample statistics.
Sampling methods
Simple random sampling may be with replacement (SRSWR) or without (SRSWOR). Stratified sampling divides the population into homogeneous strata, often with proportional allocation n_h = n·N_h/N. Systematic sampling takes a random start and then every kth unit, with k = N/n. Cluster sampling selects whole groups, and multistage sampling combines stages.
Sampling distribution of the mean
For SRSWR, V(x̄) = σ²/n. For SRSWOR, V(x̄) = (σ²/n)·(N − n)/(N − 1), where (N − n)/(N − 1) is the finite population correction. The standard error is the square root of this variance and falls as n grows. For large n the central limit theorem makes x̄ approximately normal.
Related distributions
t = (x̄ − μ)/(s/√n) with n − 1 degrees of freedom; χ² = (n − 1)s²/σ² with n − 1 degrees of freedom; F is the ratio of two independent χ² variables, each divided by its degrees of freedom.
Worked example
N = 1000, n = 100, σ = 50 under SRSWOR: SE = (50/10)·√(900/999) = 5 × 0.9492 ≈ 4.75, slightly below the with-replacement value of 5.
Traps
Omitting the finite population correction in SRSWOR; using N instead of N − 1 inside the correction factor; confusing the population distribution with the sampling distribution; mixing up the degrees of freedom of t and χ².