# Hypothesis test for binomially distributed variable with large n and small p

Example: I want to compare the number of successes in a sample of ~10,000,000 independent bernoulli trials against a known population probability of success of ~.01. Thus, expected success count for 10,000,000 trials should be 100,000.

What's the quickest hypothesis test to implement for a large n, small p? I would prefer to avoid the exact binomial test if possible as I'm trying to code an automated hypothesis test.

• @benrolls and EMS>> In this case the standard deviation under the null is 315, meaning if the null is true you'd expect about 95% of runs to give resulting in the range $1.0 \cdot 10^5 \pm 730$. The normal approximation will be fine. (In fact, if you're happy with a 5% significance level, that's your hypothesis test right there.) – Cyan Mar 16 '12 at 19:36