I'm doing an analysis to examine if my data points are clustered along a polar axis (the cycle of a repeated playback sound followed by silence). I'm interested in:
Are the data points clustered along a specific region of the playback cycle. I'm using Rayleigh's test for this.
If they are clustered around a point on the axis, what is the probability that this point lies outside a specified region of the axis (essentially, is the data centered in a region of the playback cycle in which the playback sound is not playing)
For #2, I am considering doing bootstrapped estimates of the directional mean from my resampled data. Then using the distribution of the bootstrapped means to construct a 95% confidence interval, where the 95% of the points closest to the mean from my actual data are used to define the CI.
Essentially, if the CI does not overlap the playback period of my playback cycle, I can assume that there is a tendency to avoid the playback.
I feel that my approach may not be 100% correct and that there might already be better established methods to get what I'm looking for, but I've had a lot of trouble finding examples of circular statistics which are relevant to my question.
Any ideas? I'd also be super happy to see any references for studies which have applied circular statistics to similar questions!
Edit: Ecological Relevance
So this question was in the context of playback experiments on singing birds. An area of ecology where there aren't really good statistical templates to follow in the literature (please let me know if I'm wrong here, and missed some good papers).
We had an $x$ minute track played on a loop, where the initial proportion of the cycle $p$ was some period of noise (0.5 for example), followed by silence for the remainder of the track.
I was interested if the individuals in the experiment would actively avoid singing in the noisey section of the track. So $p$ is the proportion of the track that we expect the bird to avoid singing in our alternative hypothesis, $H_1$.