Let $\{X_i\}_{i=1}^n$ be a family of i.i.d. random variables taking values in $[0,1]$, having a mean $\mu$ and variance $\sigma^2$. A simple confidence interval for the mean, using $\sigma$ whenever it is known, is given by
$$
P( | \bar X - \mu| > \varepsilon) \le \frac{\sigma^2}{n\varepsilon^2} \le\frac{1}{n \varepsilon^2} \qquad (1).
$$

Also, because $\frac{\bar X- \mu}{\sigma/\sqrt{n}}$ is asymptotically distributed as a standard normal random variable, the normal distribution is sometimes used to "construct" an approximate confidence interval.


----------


In multiple-choice answer statistics exams, I've had to use this approximation instead of $(1)$ whenever $n \geq 30$. I've always felt very uncomfortable with this (more than you can imagine), as the approximation error is not quantified.


----------


 - Why use the normal approximation rather than $(1)$?

 - I don't want, ever again, to blindly apply the rule $n \geq 30$. Are there good references that can support me in a refusal to do so and provide appropriate alternatives?


----------

**FAQ** (about the updated question.)

> What are important part of your question unanswered by any of the links?

My question is particularily about confidence intervals and is a reference request, and thefore differs from the questions in the links. It is also not answered there.


> What you mean by "appropriate alternatives"?

$(1)$ is an example of what I consider an appropriate alternative.

> In practice μ and σ are unknown, so asking for a bound on the normal approximation error is essentially useless!

Here, while $\sigma$ and $E[ |X|^3]$ are unknown, they are easily bounded. So we have a numerical bound.