Randomness condition

Every inference procedure assumes the data arrived by a chance process, and the AP course calls this the randomization condition. It comes in two versions doing two different jobs. Inference about a population needs a random sample, which is what makes the sampling distribution the right model for your statistic. Inference about a treatment effect needs random assignment of treatments to experimental units. You verify the condition by quoting the design in context, not by computing anything: "the 40 households were chosen by simple random sample from the town directory" satisfies it, and "the first 40 households on my street" does not.

What failure costs is worth seeing in numbers. Suppose 55 percent of a town supports a measure, but the one block you can reach runs 80 percent supportive. A convenience sample from that block centers $\hat{p}$ (p-hat, the sample proportion) on 0.80 whatever $n$ is. At $n = 40$ its standard deviation is $\sqrt{0.8(0.2)/40} = 0.063$; at $n = 400$ it is 0.02. The estimate gets about three times more precise around a value that is 0.25 away from the truth. More data buys a narrower interval around the wrong number.

The misreading is "I did not pick who was in the sample, so it was random." Haphazard is not random. Random means a chance device with known selection probabilities: a random number generator, a table of random digits, names drawn from a hat. Whoever happened to walk past is a convenience sample, and so is whoever chose to reply.

The two versions do not substitute for each other. Random selection buys generalization to the population and nothing about cause. Random assignment buys the causal claim and nothing about generalization, so a randomized experiment on volunteers supports cause and effect within those volunteers only. When the condition fails, say which version failed and stop there. The calculator will still return an interval, and that interval will not describe the population you wanted.