10% condition

Sampling without replacement makes the draws dependent: taking one person out changes what is left for the next draw. The standard deviation formulas $\frac{\sigma}{\sqrt{n}}$ and $\sqrt{\frac{p(1-p)}{n}}$ are built for independent draws, and the exact spread under sampling without replacement carries an extra factor $\sqrt{\frac{N-n}{N-1}}$, where $N$ is the population size. The 10% condition, $n \le 0.10N$, is the promise that this factor is close enough to 1 to ignore.

Put $n$ right at the ceiling and the size of the approximation becomes visible. With $N = 1{,}000$ and $n = 100$, the factor is $\sqrt{\frac{900}{999}} = 0.9492$, so the true standard deviation is about 5 percent below what the formula reports. The error runs in the safe direction, since a spread that is too large gives intervals slightly too wide rather than too narrow. Sample half the population, $n = 500$, and the factor falls to 0.7075: the true standard deviation is now 29 percent below the formula, which is no longer ignorable.

The sentence to correct is this one: "the sample has to be at least 10 percent of the population." It is backwards twice. The condition is a ceiling, not a floor, and it has nothing to do with being representative. A random sample of 1,000 from 300 million is about 0.0003 percent of its population and is perfectly sound, because representativeness comes from random selection rather than from the fraction sampled.

Keep the checks apart, because they do different jobs. Randomness comes from the design. Large Counts is about shape, whether a normal model fits the sampling distribution. This one is about spread, whether the independence formula is close enough to the truth. It is the check you skip when sampling is done with replacement, and the one you cite when arguing independence from a design.