A random sample of 400 is enough to give a good estimate,
regardless of the size of the population.
This is a non-intuitive result from statistics. The standard deviation - the average expected error - does not actually depend on the population size. It could be 7 billion or 7 hundred. The standard deviation only depends on the sample size, and is O(1/sqrt(n)).
In this case, with n=400, we have O(0.05). If we are testing a random variable with two values, then the true deviation is less than 0.5, so the expected error is 0.025 - we expect no more than 2.5% of mistake. That's very good!
(The bigger question is whether the sample is random or not.)
