Determining unusual binomial results from a sub-population If I have a population of 1948 students with a success rate of .154 (301 successes) and I break the 1948 students into their respective majors, how to I determine if the sub-population success rate is statistically significant? 
For example, I have a sub-population of 188 students (from the 1948 above) and I know their success rate is .2128. How do I determine whether the results from this population is significant or not? 
 A: If what you have are truly populations and subpopulations, the word "significance" in the sense of "significance test" is meaningless. The purpose of signifcance testing is to make probability-based inference about unobservable population quantities on the basis that (say) the population was randomly sampled, or randomly allocated to treatments (or whatever else the basis for the probability model is) - to use probabilistic reasoning to try to answer a question like "do the population proportions differ?".
If you actually have the population (as you state), the population quantities are not unobserved, but known, without error. You can determine if they differ by simple examination - if they are not identical, then they differ (e.g. 0.430 is greater than 0.429, even though that may not be an important difference). That is, you can immediately see the answer to the question that statistical inference is attempting to infer.
In order for probabilistic reasoning to take meaning, you'd have to make some kind of sampling assumption about drawing from some even larger population ... but then you don't have the population about which you wish to make inferences and the claim that you did is false.
So either drop the claim that you have a population (in which case you're stuck trying to explain how what you do have could be seen as a sample about which you could make a probabilistic argument), or drop any attempt at probabilistic inference and simply compare the proportions directly, focusing instead on the practical importance of any differences.
A: If you only want to know if the proportion .2128 (40 / 188) differs from the value .154 (taken as set in stone), this is a simple binomial test.  Here's an example in R:  
binom.test(x=40, n=188, p=.154, alternative="greater")

#   Exact binomial test
# 
# data:  40 and 188
# number of successes = 40, number of trials = 188,
# p-value = 0.01962
# alternative hypothesis: true probability of success is greater than 0.154
# 95 percent confidence interval:
#  0.1646788 1.0000000
# sample estimates:
# probability of success 
#               0.212766

