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I want to calculate the variance of $$\sqrt{n}\arcsin \sqrt{P}$$ and I believe I'm supposed to use the Taylor approximation where $$\sqrt{n}\arcsin \sqrt{P} = Z,$$ where $$Z = g(P).$$ I'm a bit confused though as to how to continue this. When I use the variance approximation, I get that $\operatorname{var}(Z) = \operatorname{var}(P)$ which seems wrong. I'd appreciate any clarifications/how to work through this problem.

This is where $P$ is the proportion $Y/n$ and $Y\sim Bin(n,p)$.

whuber
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Sophia
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    Please add the `self-study` tag and relate to the original problem. The question is unclear as stated. Note that $Z$ is taking a finite number of values, hence the Taylor approximation does not look particularly relevant. – Xi'an Mar 25 '20 at 07:16
  • You may find [the results of this site search](https://stats.stackexchange.com/search?q=+arcsin+proportion) to be relevant and useful. My post at https://stats.stackexchange.com/a/251661/919 gives a detailed explanation of how the arcsin arises with Binomial data, which therefore might suggest some approaches for you to take. – whuber Mar 25 '20 at 19:07

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