Sampling Distributions for Sample Proportions

For a simple random sample of size $n$, the sampling distribution of $\hat{p}$ has $\mu_{\hat{p}} = p$ and $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, and it is approximately normal when the counts are large. Three conditions carry three separate jobs: random makes $\hat{p}$ unbiased and licenses generalization, the 10% condition keeps the standard-deviation formula accurate under sampling without replacement, and $np \ge 10$ with $n(1-p) \ge 10$ delivers the normal shape. With $p = 0.60$ and $n = 100$, $\sigma_{\hat{p}} \approx 0.049$ and $P(\hat{p} > 0.65) \approx 0.154$.

The standard error gets rebuilt from the wrong parts: dividing by $n$ rather than $\sqrt{n}$, leaving $n$ out of the radical entirely, or claiming that doubling $n$ halves the spread when it takes a factor of four. And the conditions get recited rather than checked: $n \ge 30$ offered for a proportion where the large-counts condition belongs, the 10% condition cited as though it established randomness, and condition names written with none of the study's numbers attached. Each condition protects a different property, so the wrong check leaves a specific part of the answer broken.