Using binomial vs central limit theorem in hypothesis testing This is the question that I was working on:
"An airline claims that the proportion of luggage that is lost is less than or equal to 0.06. A random sample of size 200 is taken. Out of the 200 independent observations, 23 pieces of luggage were lost. Test whether the airline's claim is true at a 1% significance level."
My approach was to model the number of lost luggages as X~Bin(200, p) and use binomial distribution to find the p-value.
However, the answer used the fact that sample proportions are normally distributed (by CLT) to find the p-value.
Would both approaches be correct, and if so, which one is "better"?
 A: This is probably a case of asking your professor which convention he expects you to use in the exam and then obeying. It's not worth losing a grade for trying to be smarter than the professor. Not for such a benign matter anyway. 
If hypothetically your professor expected you to answer that p-values express direct probabilities that the hypothesis is true or a similarly colossal mistake, then I wouldn't only advice to obey but mostly to look for a better professor.
Know for the future that it is not a mistake to use binomial distributions with very high $n$. (Under the condition that the trials are independent and have the same $p$ etc. But similar conditions would apply to the normal approximation anyway.) The approximation was invented for reasons of computational complexity that really don't matter in the 21st century anymore.
It is still interesting to understand how the central limit theorem allows you to approximate a binomial distribution by a normal distribution because the central limit theorem has other domains of application unrelated to computational complexity. This is probably the reason your professor taught that approximation.
