[jason1990]

[ QUOTE ]
How different is the confidence interval which uses the estimated SD from the confidence interval which uses the true SD? Well, with 30 observations, it's not going to be much different.

[/ QUOTE ]
Okay, this was said in a sloppy, hand-wavy, sweep-it-under-the-rug type way, and, as it stands, it's probably not even true. So I feel compelled to elaborate on where the 30 comes from.

Suppose I play n sessions which, for simplicity, are all the same length, say 100 hands. Let X_i be the number of BBs I won in the i-th session and I will assume each X_i is normal with (unknown) mean w and (unknown) variance s^2. So my true standard deviation is s. Let W and S be my estimated (or observed) winrate and standard deviation, respectively.

If I knew s, then I could build a confidence interval for w by using the fact that (W - w)/(s/sqrt{n}) is normal. But I don't know s, so I must use S. And the problem is that (W - w)/(S/sqrt{n}) is not normal. But when n is at least 30, then this quantity is close to normal, so in this sense we are justified in using S instead of s in our usual construction of the confidence interval. But that does not mean that the numerical values of s and S are necessarily close.

Which means we may not be justified in replacing s with S in other formulas, such as the RoR formula.