First, I wrote an incorrect formula for the normal distribution. Please see the text for the correct formula. You may wish to use this formula for Project 4, but you will not be tested on it in the final exam.
The second error was in my description of the Central Limit Theorem. I mistakenly stated that with enough tests (N), the output will follow a normal distribution with variance sigma^2 and the average of the output will follow a normal distribution with variance (sigma^2)/N.
Actually, only the average of the output will follow a normal distribution. The output values will most likely follow a different distribution. For example, consider the simple function of multiplying a uniform random variable by 2. We would certainly not expect this output to follow a normal distribution.
For the Central Limit Theorem, we can compute a new random variable Z_N which is based on the output of the function, the number of tests, N, the true mean and the true variance, sigma^2. The Central Limit Theorem states that Z_N tends to follow a normal distribution as N gets sufficiently large. From the Z_N distribution, we can prove the key point of the theorem which is that the average of the output follows a standard distribution with variance (sigma^2)/N.