So for a large n, we have , where Z ~ N(0,1)
But we let (it's weird that you're given n=15, let's just ignore it)
recall that and it gives
But now recall that the arcsine function is strictly increasing. So it's bijective. So
From previous parts of a rather long question, I have shown that converges in distribution to the standard normal distribution. The next part of the question is:
Use this fact to derive an approximate condence interval for known as the arcsine-interval, and compute this interval for some arbitrary confidence level, .
It's just algebra, but I'm totally unsure about how to manipulate all these arcsines all over the place.