How to estimate population proportions for colored beads in a jar Suppose I have a jar of colored beads. The jar contains about 15000 of those beads and they come in 10 different colors. I want to know whether there is a a difference between the true population proportions for each color.
I sampled randomly from the jar four times using a small cup (approx. 120 beads total per cup) and counted the number of colors. Each sample consisted approximately of the same number beads (I always filled the cup to the top).
One way of analyzing this dataset is using a Poisson GLM (using R syntax):
glm(count ~ color, family = "poisson")

with count being my response variable and color my predictor.
However, this will give me the average color count per cup (or rate) if I am not mistaken. But instead I would like the parameter estimates of the population proportions of colors in the jar.
Since I have more than two colored outcomes, I believe one way to go about this is by using a multinomial model. In R, for example, this could be done using the multinom() function from the nnet package:
multinom(color ~ count)

Now color is my response variable and count my predictor.
My question is, is a multinomial model the correct approach estimating the population proportions for each color?

Following Henry's comment, I don't think the fact that you take multiple samples makes any difference. Just add them together so you have one large sample. (Given your relatively large population compared to your samples, I don't think sampling with or without replacement is going to make much difference.)
Then to test the hypothesis that the actual distribution is uniform (i.e. all colors have the same proportion) just run a Chi-square goodness-of-fit test.
