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.


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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.


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 - 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?


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**Answering Glen's concerns.**

> 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. 

> this is one of the difficulties with asking shopping-lists of questions

I precised my question, thank you. 

> 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 like that is essentially useless

Here, while $\mu$ and $\sigma$ are unknown, they are easily bounded. So we have a numerical bound.