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.