Larger sample vs smaller sample odds Say we have two hospitals, hospital A and hospital B. In hospital A there were 230 births and in hospital B there were 560. If the odds of giving birth to a boy is 51%, then which hospital is most likely to have 55% of the births to be boys? Guess: My guess is hospital A, because the sample size is smaller -> standard error is larger so most likely to diverge further from 51%.

Couldn't you solve this by using binomial distribution?

* In hospital A, the number of boys follows a binomial distribution with n = 230 and p = 0.51

* In hospital B, the number of boys follows a binomial distribution with n = 560 and p = 0.51




If you know the formula for binomial distribution, you can just plug in the values and compute:

* P(55% boys in hospital A) = ... ?

* P(55% boys in hospital B) = ... ?




I haven't done the computations myself, but intuitively I would say that your final paragraph makes sense. The larger the sample size is, the closer we expect the real mean and the sample mean to be. To have a sample mean of 55%, we need more variation, since this is different from the 51% that we expect the sample mean to converge to.