# 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"?

• What's your criterion for "better"? Sep 20 '17 at 3:05
• I think you should add the self study tag. The normal approach is an approximation though probably reasonably good with 200 observations. Using the binomial is an exact test. Sep 20 '17 at 3:19
• Well it depends whether the assumptions underlying the binomial are fulfilled (independence and same p) and how Well the normal approximates the binomial with n=200 and p=0.06 but if you compute the p value using the binomial ( and you do it correctly) then it makes sense
– user83346
Sep 20 '17 at 4:25
• @Michael The Normal approximation is problematic here because the expected count is only 12. Its p-value is less than one-third the correct one (although both are small).
– whuber
Sep 20 '17 at 13:21

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.