Independence Condition vs Randomness 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.

Every inference procedure assumes the data came from a random sample or a randomized experiment; the AP course framework calls this the randomization condition. Without it, the sampling distribution you are leaning on does not describe your data, so the p-value or interval you compute says nothing about a larger group. Random selection is what lets you generalize to a population, and random assignment is what lets you claim cause and effect. Writing "the 40 households were chosen by simple random sample from the town directory" satisfies the condition, while "the first 40 households on my street" fails it no matter how many you add.