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I have a random variable, supposedly from a Bernoulli distribution, and I want to do some tests on its mean. I'd like to assume that the sample mean is normally distributed, but I'm not sure how reliable that assumption is. I want to estimate how such assumption would damage the power and significance of my tests. How can I do that?

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I have a random variable, supposedly from a Bernoulli distribution, and I want to do some tests on its mean. I'd like to assume that the sample mean is normally distributed, but I'm not sure how reliable that assumption is.

I assume $X_i \stackrel{_\text{iid}}{\sim} \text{Bernoulli}(p)\,,\quad i=1,\ldots,n$

If the Bernoulli parameter $p$ is not close to $0$ or $1$, a normal approximation to $\bar(X)$ can work for many purposes with $n$ moderately small; in some situations n=20 can be reasonable ... but if $p$ is small (or large) it may need to be a lot bigger.

A common "rule of thumb" (which you should regard with suitable caution) used to decide whether $n$ is large enough include $\min(np,n(1-p))>5$. I've also seen $np(1-p)>9$. There are a number of other rules - see the discussion here:

https://en.wikipedia.org/wiki/Binomial_distribution#Normal_approximation

Such rules may be a reasonable starting point, but their utility varies across the particulars of the application (and user's preference functions).

I want to estimate how such assumption would damage the power and significance of my tests. How can I do that?

Simulation can be very useful; choose some reasonable set of $p$'s and $n$'s. Note that only values either above or below $\frac{_1}{^2}$ need be considered, since the closeness to normality will be the same for $p=p_0$ and $p=1-p_0$), and keep in mind that the approximation will be worst the further $p$ is from $\frac{_1}{^2}$.

This way you can look at the impact on the specific procedures you're carrying out, and for the particularly quantities of interest -- whether or not the impact of the approximation could be calculated algebraically.

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