Q.A retail chain has 800 employees, spread across 40 branches with 20 employees at each branch; the branches are further grouped into 4 regional zones of equal size (10 branches, i.e. 200 employees, per zone). HR wants a sample of 80 employees for a job-satisfaction survey. Explain, with a suitable illustration, how the sample of 80 could be drawn using

Each method applies the same underlying idea — giving every employee a known, non-zero chance of selection — but structures the selection differently around the population's zone-and-branch grouping.

1. Simple Random Sampling: number all 800 employees from 1 to 800, and use a random-number table (or the lottery method) to draw 80 numbers. Every employee has an equal, independent $\dfrac{80}{800} = \dfrac{1}{10}$ chance of selection, regardless of branch or zone.
2. Stratified Sampling (by zone): treat each of the 4 zones (200 employees each) as a stratum. Since all four zones are of equal size, the sample of 80 is split equally: $\dfrac{80}{4} = 20$ employees drawn at random from each zone. This guarantees every zone is represented in the final sample.
3. Systematic Sampling: arrange all 800 employees in a list (e.g., by employee code) and compute $k = \dfrac{800}{80} = 10$. Choose a random start between 1 and 10 (say, 6), then select employees numbered 6, 16, 26, 36, ..., up to 796 — exactly 80 employees.
4. Cluster Sampling (by branch): treat each of the 40 branches (20 employees each) as a cluster. Since $4 \times 20 = 80$, randomly select 4 of the 40 branches and survey every employee at those 4 branches — no employee is drawn from any of the other 36 branches at all.
   ✓Final answer

   (i) SRS: number all 800, randomly draw 80. (ii) Stratified: 20 employees drawn from each of the 4 zones (200 each). (iii) Systematic: k=800/80=10, random start, then every 10th employee. (iv) Cluster: randomly select 4 of the 40 branches (4x20=80 employees) and survey everyone in those branches only.

A common confusion is mixing up Stratified and Cluster sampling — in stratified sampling a partial sample is drawn from EVERY group, while in cluster sampling entire groups are skipped and only a few chosen clusters are surveyed in full. Also, cluster sampling only gives an exact target sample size when the cluster size divides evenly into it, as engineered here (4x20=80).