# Hypothesis testing on two proportions in two finite populations

I have data collected over two years where:

• Year 1 = 34 successes in the representative sample of 49 taken from the finite population of 122, and

• Year 2 = 45 successes in the representative sample of 68 taken from the finite population of 175.

What test do I use to see if there is any significant difference between the two?

• You'll want to use a two sample test for difference in proportions: onlinecourses.science.psu.edu/stat414/node/268
– mai
Commented Mar 12, 2018 at 2:56
• Thank mchen. This test doesn't take into account the statistically representative sample size of the total population from within which the successes were achieved - can you test the successes on the total population without needing to consider the sample size within which they were achieved? Commented Mar 12, 2018 at 3:24
• Would simulation be a reasonable answer? 34 successes in 49 gives you a distribution of credible values for the rate $\theta_1$ in the first year. so credible values for the successes in the population in year 1 are 49 plus binomial samples taken from 122 - 49 = 73 individuals. This can easily be simulated. The same can be done for Year 2 and then you could compare. That is, if simulation is suitable for the problem behind the scenes. Commented Mar 12, 2018 at 13:37
• @Bernhard I am with you on this. I think that a parametric bootstrap approach is warranted. A random sample is drawn as described. In group 1, 34/15 of the 122 are definitive successes/failures and the remaining 73 are Bernoulli according to $p=34/49$. In group 2 similarly. I note you call them credible intervals, and as an ardent frequentist, am struggling to justify a frequentist approach. I might agree it's a fundamentally Bayesian hypothesis and approach. Commented Apr 13, 2018 at 14:32