Independence condition

Independence is what lets you multiply probabilities and use $\sigma/\sqrt{n}$ (sigma over the square root of n) or $\sqrt{p(1-p)/n}$ for the spread of a sampling distribution. You almost never verify it directly; you argue for it from the design, since random selection with replacement makes it exact and sampling without replacement makes it close enough when $n \le 0.10N$ (n at most 10% of the population size N). Interviewing 50 randomly chosen shoppers from a list of 5,000 qualifies, because 50 sits well under $0.10(5000) = 500$. In an experiment, random assignment to treatments is what supports treating the groups as independent.