Quick questions on Random sampling and the distribution of sample proportions (TCE Mathematics Methods, Tasmania)

Imagine taking many independent random samples of the same size $n$ and recording $\hat{p}$ each time. The collection of those values has its own distribution, called the sampling distribution of the sample proportion. Because the count of successes is binomial, dividing by $n$ rescales it, and the centre and spread follow directly.

A useful way to understand this is simulation. If you repeatedly generate samples from a population with a known $p$ and plot all the resulting $\hat{p}$ values in a histogram, you see a roughly bell-shaped cluster centred on $p$. Larger samples produce a tighter, taller cluster, visually confirming that the standard error falls as $n$ rises.