Sampling Theory and Sampling Distributions
What the domain covers
This domain tests census versus sample, probability and non-probability sampling, sampling and non-sampling error, standard error, the sampling distribution of the mean and the role of t, chi-square and F distributions. The exam expects correct interpretation of standard error and degrees of freedom.
Core rules
- Probability sampling: simple random, stratified, systematic, cluster. Non-probability: convenience, judgement, quota.
- Sampling error arises because a sample is used; non-sampling error includes measurement, non-response and recording mistakes.
- Standard error of the mean: SE = σ/√n when σ is known; use s/√n when estimated. For sampling without replacement from a finite population, multiply by √[(N − n)/(N − 1)].
- The sampling distribution of x̄ tends to normality as n grows (central limit theorem), with mean μ and SD σ/√n.
- t is used for means with unknown σ and small n; chi-square for variance and goodness of fit; F for comparing variances.
Worked example
A population has σ = 10 and a sample of n = 25. SE = 10/√25 = 2. If sampling is without replacement from N = 100, the finite population correction is √[(100 − 25)/(100 − 1)] = √(75/99) ≈ 0.870, giving SE ≈ 1.74. Increasing n to 100 would halve the simple SE to 1.
Traps
Sampling error is not a blunder; it is unavoidable random variation. Do not apply the finite population correction when sampling with replacement. SE decreases with √n, not linearly with n. Larger samples reduce sampling error but may not reduce non-sampling error. Check degrees of freedom before using t, chi-square or F tables.