Conditions for the Normal Approximation

A sample proportion's distribution is only approximately normal when successes and failures are both common enough to spread the distribution out symmetrically. When p is near 0 or 1, like p = 0.02, failures (or successes) pile up right at the boundary of 0, so the distribution of sample proportions gets squeezed and skewed instead of bell-shaped. There just isn't room for it to spread evenly on both sides.

The large counts condition, np ≥ 10 and n(1-p) ≥ 10, checks that both outcomes show up often enough to let the shape fill in and look normal. The 10% condition, that n is no more than 10% of the population, protects a separate assumption: independence between observations when sampling without replacement. One condition is about shape, the other is about independence. Both have to hold before you can treat the sampling distribution as normal and attach z-based margins of error or p-values to it.

This sits right before you build confidence intervals and significance tests for a single proportion, which both lean on that normal shape to find critical values and p-values. If the conditions fail, the whole machinery of those later lessons, the standard error formula, the z-scores, the interpretation, rests on a shape assumption that doesn't actually hold, so checking conditions always comes first, not as an afterthought.