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

Here, while $\sigma$ and $E[ |X|^3]$ are unknown, they are easily bounded.

Please note that my question is a reference request particularly about confidence intervals and therefore is distinct from the differs from the questions that were suggested as partial duplicates here and here. It is not answered there.

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

Here, while $\sigma$ and $E[ |X|^3]$ are unknown, they are easily bounded.

Please note that my question is a reference request particularly about confidence intervals and therefore is distinct from the differs from the questions that were suggested as partial duplicates here and here. It is not answered there.

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

Note that here, while $\sigma$ and $E[ |X|^3]$ are unknown, they are easily bounded. So we have numerical bounds we can obtain.

Please note that my question is a reference request particularly about confidence intervals and therefore is distinct from the differs from the questions that were suggested as partial duplicates here and here. It is not answered there.

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

Here, while $\sigma$ and $E[ |X|^3]$ are unknown, they are easily bounded.

Please note that my question is a reference request particularly about confidence intervals and therefore is distinct from the differs from the questions that were suggested as partial duplicates here and here. It is not answered there.