(very basic) One-sample test for binary data I've repeatedly measured a continuous variable and each measure has been assigned a populational percentile range it falls into (percentile ranges were estimated for general population in another study). The exemplary barplot depicts the share of each group in my sample during different time points.
I'd like to test, whether the percentage, that specific range constitutes in certain time point (eg. share of 90-100th percentile in the 6th time point) differs between my group and general population.
The simplest way would be to perform a one-sample t-test (eg. against the mean of 0.1 in case of 90-100th percentile, as 10% of values will fall above 90th percentile in the population). But is there any alternative if my sample gets small? 
It would be ideal to perform Fisher's exact test, but I have no reference group - I can only assume, that in general population 1/10 samples will fall into that percentile range.

 A: Ok, given mjktfw's comment, I think I have at least something of an answer:
1) You say you turned a continuous variable into one of 5 percentile ranges; I would not do this. Categorizing a continuous variable loses information.
2) If you have some really good reason to do this, OK.
3) You say 

I'd like to test, whether the percentage, that specific range
  constitutes in certain time point (eg. share of 90-100th percentile in
  the 6th time point) differs between my group and general population.

Since you are assuming that 10% of the general population was in 90-100, you could just do a one way chi-square to see if your percentage is substantially different from 10%. E.g. if you had 20 people in the top 10% and 80 in the bottom 90 % you could use R:
chisq.test(x = c(20,80),p = c(.10,.90))

If you wanted a graph of this, a mosaic plot would work well.
Then you give a graph of all 5 percentile categories over a series of time points, presumably to be able to see how the proportion in each category changed over time.  Instead, I would make a line plot with 5 lines: One for each category; time would still be on the x-axis (and I'll presume the times were equally spaced); the y axis would be proportion. 
If you  want to test more complex hypotheses, please state what they are.
