# 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?

• What would it mean for that sub-population to be "significant" in your situation? – gung - Reinstate Monica Dec 5 '14 at 18:56
• @gung Would it be safe to use the common 2 standard deviations from the mean? If so, which series of mean and standard deviations to I use, the full population or the sub-population? – Andy Levesque Dec 5 '14 at 19:03
• I'm using the standard definition of "mean" here, not as average. What would it mean for these to be "significant"? What would that imply about the world? What is your substantive question? – gung - Reinstate Monica Dec 5 '14 at 19:05
• @gung Sorry, that's what I assumed you meant, but then I threw in the mathematical 'mean' in my response. We don't have a perspective of what is significant going into this review, so it would be best to use the statistical definition of 'significant', target those populations first, and then further define significant after the initial review. – Andy Levesque Dec 5 '14 at 19:56
• What would it mean in terms of the way the world works? What is the hypothesis you want to test? Saying something is 'significant' only means something in the context of a particular hypothesis being tested. 'Testing to see if it's significant' in a vacuum is an empty phrase. What if I asked you 'Is my right thumb significant?' or 'Is my left pinky significant?', you would probably reply that the question is empty & cannot be answered. In your case, what would it mean for that sub-population to be "significant"? – gung - Reinstate Monica Dec 5 '14 at 20:15

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.

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