This is a fun analysis to see. Here's a suggestion for cleaning up the stats, though it won't help you get around people noting that there's a selection bias for the poll results themselves.
If you're interested in showing that the frequency distribution for personality types is different on Hacker News than in the general population, then the right way to show this would be to run a single multinomial test to see whether your empirical multinomial distribution is outside of the range of chance variation from the canonical distribution listed on Wikipedia for the MBTI. See http://en.wikipedia.org/wiki/Multinomial_test for a description of this hypothesis test.
To give you a sense of why your multiple comparisons across binomial tests is problematic, imagine that the INTP group is underrepresented because all of the people who would test as INTP tested as ENTP this time around; in this case, you run two different binomial tests, but you really only have one result, because the size divergence in one group is effectively the cause of the size divergence in the other group.
I hope that makes it clear why you should really treat your data set as containing an N of 1 and doing a simple null hypothesis test of the probability of getting your results given the canonical values for the global multinomial distribution, rather than testing based on a series of binomial marginal distributions.
Comments
This is a fun analysis to see. Here's a suggestion for cleaning up the stats, though it won't help you get around people noting that there's a selection bias for the poll results themselves.
If you're interested in showing that the frequency distribution for personality types is different on Hacker News than in the general population, then the right way to show this would be to run a single multinomial test to see whether your empirical multinomial distribution is outside of the range of chance variation from the canonical distribution listed on Wikipedia for the MBTI. See http://en.wikipedia.org/wiki/Multinomial_test for a description of this hypothesis test.
To give you a sense of why your multiple comparisons across binomial tests is problematic, imagine that the INTP group is underrepresented because all of the people who would test as INTP tested as ENTP this time around; in this case, you run two different binomial tests, but you really only have one result, because the size divergence in one group is effectively the cause of the size divergence in the other group.
I hope that makes it clear why you should really treat your data set as containing an N of 1 and doing a simple null hypothesis test of the probability of getting your results given the canonical values for the global multinomial distribution, rather than testing based on a series of binomial marginal distributions.