$X$ has a pdf $f(x) = c \sin x$ on $[0, \pi]$. Assuming $n$ is large, find the probability that the sample mean, $\overline{X}_n$ is less than $1$.
I found the mean $\pi/2$ and am trying to find $Z$ but I am not sure if I need use normal distribution.
My solution :
$$Z=\frac{\overline{X}_n-\mu}{(\sigma/\sqrt{n})}=\frac{1-\frac{\pi}{2}}{(\sqrt{\pi^2-8})/2}=\frac{1-\pi}{\sqrt{\pi^2-8}}\approx-1.566$$
Next,
$$\mathbb{P}(\overline{X}_n<1)\approx\mathbb{P}(Z\le-1.566)\approx0.0594$$