Pharma Tablets are produced in large batches. Individual tablets have weight that is uniformly distributed between $22$ and $24$. A sample of $100$ tablets is randomly selected and the average is computed. Find the probability that average exceeds $23.5$.
My working:
Let $W$ be the random variable representing the weight of tablets then since $W$ is uniformly distributed its $pdf$ is given by:
$f(w)=\frac{1}{2}$ ; $\hspace{0.5cm}$ $22<w<24$
Honestly I don't know how to tackle this problem further. A bit of guidance/help will be appreciated.
$W\sim\text{Uniform}(22,24)$ and has expected value $23$ and variance $\frac 1{12}(b-a)^2=\frac 1{12}(4)=\frac 13$.
The central limit theorem has that $\bar W\sim \text{Normal}\left(23, \frac1{3(100)}\right)=\mathscr N(23, 1/300)$.
So $\Pr(\bar W>23.5)=\Pr\left(Z>\frac{23.5-23}{\sqrt{1/300}}\right)=\Pr(Z>8.660254)=2.353571\times 10^{-18}\approx0$.