Independence Condition vs Randomness Condition

Independence means one observation carries no information about another, and it is a claim about the design rather than something you can read off the data. Sampling with replacement makes it exact. Sampling without replacement makes it false, because removing one person changes the odds for the next, so the course substitutes the 10% condition, $n \le 0.10N$, as an argument that the dependence is small enough to ignore. In a randomized experiment, random assignment supplies it instead.

The reason it is load bearing is one algebraic step. Variances of independent quantities add, so the sum $X_1 + \cdots + X_n$ has variance $n\sigma^2$, and dividing that sum by $n$ divides its variance by $n^2$, leaving $\frac{n\sigma^2}{n^2} = \frac{\sigma^2}{n}$ as the variance of $\bar{x}$ (x-bar, the sample mean). Take the square root and you have $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ (sigma over the square root of n). With $\sigma = 20$ and $n = 100$, that is $20/10 = 2$. The identical argument on a sum of independent successes produces $\sqrt{\frac{p(1-p)}{n}}$ for a proportion. Take independence away and the variance of the sum picks up covariance terms, the addition step fails, and the 2 is just a wrong number.

The sentence to unlearn is "the sample was random, so the observations are independent." Those are two different claims. A simple random sample drawn without replacement is random and dependent at the same time. Randomness is what centers the statistic on the parameter; the 10% condition is what makes the spread formula usable. Mislabeling one as the other is the error the course specifically flags.

Two boundaries. When data come from a randomized experiment the course treats the 10% check as unnecessary in the two-sample and chi-square procedures, because assignment rather than selection is doing the work. And every two-sample procedure needs a second independence claim stacked on this one, that the two groups are independent of each other, which is exactly what a matched pairs design gives up on purpose so that it can pair the observations instead.

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.