Consider the Central Limit Theorem, and the Correction for Continuity that is used when dealing with a Binomial distribution (read Chapter 4.3 for full details, especially 4.3.3). Let's explore how much better' this Correction makes our estimate. For reference, we use Corollary 4.3.2 from the textbook here: if X ~Bin(n, p) and p = X/n is the proportion of successes, then (under some conditions), the sampling distribution of p is approximately normal with mean p and standard deviation √p(1-p)/n.
Consider a binomial distribution with p = 0.146. Compute the theoretical (CLT) sampling distribution for n = 32. Then, create m = 1000 replicates:
Using rbinom, sample m = 1000 random samples from a binomial distribution, size n = 32, prob= 0.146.
If you obtain any samples with X = 0, replace these with additional samples or drop them from consideration (noting that this changes your effective m!).