# Test if the sample mean is statistically significantly different from zero

Let $X_{1},\ldots,X_{N}$ be IID. The plug-in estimator for the mean is the sample mean $\bar{X}=(X_{1}+\cdots+X_{N})/N$. The standard error of this estimator is $\sqrt{\operatorname{Var}(X_{1})/N}$. Therefore, a normal confidence interval for the mean is \begin{equation} \bar{X}\pm c\sqrt{\frac{\operatorname{Var}(X_{1})}{N}}. \end{equation} In short, if we want to test if the mean is statistically significantly different from zero (assuming normality), the number of samples needs to satisfy \begin{equation} N>\frac{c^{2}\operatorname{Var}(X_{1})}{\left|\bar{X}\right|^{2}}. \end{equation}