# Accuracy in approximation with central limit theorem

I have set of random values with the same distribution $y_1, \ldots, y_N$

$$T = \frac{1}{N}\sum_{j = 1}^{N} y_j$$

I want to find the confidence interval for the mathematical expectation $E(Y)$

I can use an approximation:

$T∼N(μ,\frac{\sigma^2}{N})$, where $\sigma=\frac{\sum^N_{i=1}(y_i−T)^2}{N−1}$.

This approximation is valid asymptotically. Then my confidence interval is $T±1.96 \frac{\sigma}{\sqrt{N}}$.

Of course, this method has a mistake, because I only have an approximate Normal distribution and must use the sample variance.

How can I deal with this inaccuracy and make a final confidence interval?

• Replacing $\sigma^2$ by the sample variance is a good approximation if n is large. The convergence rate relies on additional assumptions over moments of higher order. You can also obtain a nonparametric bootstrap interval for the mean by resampling with replacement and obtaining the corresponding means and compare it with the CLT approximation. – user10525 May 27 '12 at 8:49