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.


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