Sampling Techniques and Design of Experiments
GATE ST tests how data are selected and how experiments are structured. Sampling questions emphasize unbiased estimation and standard errors; design questions emphasize randomization, blocking, and valid ANOVA comparisons.
Core rules
- In simple random sampling without replacement (SRSWOR), the sample mean is unbiased and Var(ȳ)=(1−f)S²/n, where f=n/N is the finite population correction.
- Stratified sampling reduces variance when strata are internally homogeneous. Proportional allocation uses nh/n=Nh/N.
- Cluster sampling selects groups, often increasing variance relative to SRS because units within clusters are correlated.
- Horvitz–Thompson estimation weights observations by inverse inclusion probabilities.
- In designed experiments, CRD suits homogeneous units; RBD removes block effects; factorial designs estimate main effects and interactions. Confounding trades high-order interactions for fewer runs.
Worked example
A population has N=100 and a sample of n=10 is taken by SRSWOR. If S²=25, then Var(ȳ)=(1−10/100)(25/10)=0.9×2.5=2.25, so the standard error is 1.5. Ignoring the finite population correction would give √(25/10)=1.581, which overstates uncertainty. The correction matters most when n is a large fraction of N.
Traps
Do not use σ²/n without the finite population correction in SRSWOR. Do not treat a systematic sample as automatically equivalent to SRS; periodicity can bias it. In ANOVA, confounding must be planned, not inferred after looking at significant effects. Replication is not the same as repeated measurement on one unit.