# Test for difference in proportions with small probabilities

I want to perform a test for differences between 2 binomial populations. The probabilities are small (usually less than 10 %). I can define the successes as "rare events".

I think that the z-test for comparing means is not suitable since the binomial distribution is a lot different than the normal distribution in case of extremely low probabilities. What is the best test to use?

• How many observations did you use for computing each proportion ? i.e. what are the denominators ?
– user83346
Aug 19, 2015 at 8:11
• It depends. Usually the number of observation is big (minimum 1000). The number of successes is relatively small and this is the reason the proportion is small. Thanks (By the way this is Tal G) Aug 19, 2015 at 8:27
• Well if the number is large enough, I think that you can work with the normal approximation for the Binomial distribtion, for small Binomials $X_1$ and $X_2$, you can compute the exact distribution of $X_1 - X_2$ (or even another function than '-') and then use this distribution to do the hypothesis test (assuming independence is a reasonable assumption)
– user83346
Aug 19, 2015 at 9:28
• Please register & merge your accounts (you can find out how in the My Account section of our help center), then you will be able to comment on & edit your own question. Aug 19, 2015 at 9:34
• Did you happen to check link I think it will answer the question (?) Aug 19, 2015 at 14:26