# Normal approximation of sum of uniform independent RVs using CLT

Let $$X_1$$, $$X_2$$, ... $$X_{16}$$ and $$Y_1$$, $$Y_2$$, ... $$Y_{16}$$ be independent uniform random variables over the interval [-1,1] and let:

$$W = \frac{(X_1 + .... + X_{16}) + (Y_1 + .... + Y_{16})}{16}$$

Use the central limit theorem to find a normal approximation to the quantity:

$$P(|W-E[W]| < 0.01)$$

So if we're using $$P(S_n < c) = \phi(z)$$ where $$z = \frac{c- n\mu}{\sqrt{n}\sigma}$$ then we have:

$$z = \frac{0.01}{4 \sqrt{\frac{1}{3}}} \approx \phi(4.33\times10^-3)$$

But then why is the answer I have from class:

That we approximate using a normal RV $$N\text{~}(0,\frac{1}{12})$$ and so it's:

$$\approx 1- P(|z|>0.01\sqrt{24})$$

that doesn't make sense to me.