Question: Problem:

I have 200k data samples which are class imblanced (10% positive class, 90% negative class). I split the data in exactly half so my training set is 100k samples and my test set os 100k samples. I train algorithms A and B. 

Algorithm A discriminates between the two classes using the test set and it achieves an AUC of AUC\_A. Algorithm B, which is an improvement on A, gets AUC we call AUC\_B. I want to determine if AUC\_B > AUC\_A by chance or not (statistically significant).

What is an algorithm to determine this answer? (say we set p<0.001)

Answer: 
... because the mean of the $n$ values is not $\mu$.

A sample mean would normally be denoted as $\bar{x}$; it's not the case that $\bar{x}=μ$ in general. Consider the case where I have 99 tickets labelled 1,2,...,99. The population average of those values, $μ$, is 50. I draw five of them and get 76, 48, 34, 8, 71, which averages 47.4. If we take the knowledge that the population mean is 50 and the first four sample values were 76, 48, 34, 8 that doesn't really constrain that last value (if we're sampling without replacement there's a few values it can't be, but otherwise it could be any of the other values and so could produce a wide variety of sample means).
Can't help you with a formal answer but I have been thinking about this topic.

I guess that it is about the concept of a restrain.

When you observe the values of a population, if you calculate the population mean you are not actually setting anny restrain, just describing the population with a parameter that happens to be the population mean.

You do eliminate a degree of freedom with the estimation of the sample mean because you force that your population of $n$ to have a sample of $n\_s$ such that

$$\sum\_{i=1}^{n\_s} (x\_i-\bar x)=0.$$

This is the way I see that; if someone can bring a more complete answer I'd like to read it.