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.