Approximate th Probability of a Sum of 16 Independent Uniform R.V.s This question has to do with the Central Limit Theorem, uniform random variables, and cumulative distribution functions, I believe, but I'm not quite sure how to apply them all in the proper way.
Q: Approximate the probability that the sum of 16 independent uniform (0, 1) random variables exceeds 10.
Sample size = 10
I suppose I want P( Z > 10), where we use a CDF here.
In my class I was given an example of this, but in that example we were given the standard deviation (which was divided by the square root of the sample size). However, I'm not sure how to apply it in this case. A google search didn't give me the answer I wanted, so I'm hoping someone here can clarify.
TL;DR: How can I apply the central limit theorem here to solve this particular problem? 
 A: Not sure if this was ever totally resolved. It is a question
that raises a number of basic issues.
Following the terminology of @mattbiesecker,
Let $Y = X_1 + X_2 + \cdots X_{16},$ where $X_i$ are
a random sample from $Unif(0,1).$
Then
$$E(Y) = E(X_1) + E(X_2) + \cdots + E(X_{16}) = 16(1/2) = 8$$
and, by independence,
$$V(Y) = V(X_1) + V(X_2) + \cdots + V(X_{16}) = 16(1/12) = 4/3.$$
Then, by the CLT, $Y$ is very nearly $N(\mu=8,\sigma^2=4/3),$
and so $\frac{Y - 8}{\sqrt{4/3}}$ is approximately standard normal.
Thus 
$$P(Y > 10) \approx P\left(Z = \frac{Y-8}{\sqrt{4/3}} > \frac{10-8}{\sqrt{4/3}} = \sqrt{3}\right),$$
where $Z \sim N(0,1),$ standard normal.
Finally, from normal tables or software, 
$$P(Z > \sqrt{3}) = 1 - P(Z \le \sqrt{3}) \approx  0.0416.$$
Notes: 
(1) For a random variable $Y,$ mean and expectation are alternate
terminologies for $E(Y).$ 
(2) The approximation in your question is quite good. Before modern methods using
transcendal functions were computationally convenient, it was customary
to simulate a random standard normal variable by adding 12 independent standard uniform random variables and subtracting 6.
[Pseudorandom generators usually produce results 'practically' indistinguishable from random $Unif(0,1).$]
(3) In a million simulated performances of your exact experiment (adding 16 independent
standard uniforms), the fraction of sums exceeding 10 was  0.041369.
Unlike the CLT, approximation by simulation can be done to almost any
desired degree of accuracy by increasing the number of iterations.
(4) Generally, in 'favorable circumstances' (where a moderate number of independent random variables
from a symmetrical distribution without 'fat' tails are summed or averaged), one
can expect 2 or 3-place accuracy from an approximation based on the CLT.
(5) If any part of the main derivation above these 'notes' remains unclear, please leave a specific Comment below, and I (or someone else) will try to help.
