This is a followup question to Standard error for the mean of a sample of binomial random variables
The previous answer mentions that the standard error of the sampling distribution is sqrt(kpq/n), where k is the number of trials in each binomial experiment, p is the probability of success, q is (1-p) and n is the number of experiments in generating the sampling distribution.
For the normal approximation of the binomial confidence interval, the standard error is sqrt(pq/n). Does that mean the normal approximation is achieving this approximation by reducing the binomial experiments to bernoulli trials (k=1)?
If I had a case where each binomial experiment had 25 trials and 100 experiments were run (k=25, n=100), would the standard error used for the binomial confidence interval be sqrt(pq/4)?