Can you calculate confidence intervals for the population mean if the obtained sample is not normally distributed?

Question

From my knowledge, if the obtained sample is approximately normally distributed, we can use t-tables to calculate the population mean confidence interval without knowing the population standard deviation.
However is there a way to calculate the population mean confidence interval if the obtained sample is not normally distributed??

Answer 1

The answer is yes.

First, in many cases, the sample is not normally distributed but you can use the central limit theorem to make a normal approximation and obtain an asymptotical confidence interval.

But you can also find exact confidence intervals even if the data is not normal. You need to find a pivotal quantity, i.e. a statistic whose distribution does not depend on the parameter of the underlying distribution.

For example let $X_1,\dots,X_n\sim\text{Exp}(\lambda)$. We have $\lambda\bar{X}\sim \Gamma(n,n)$. Thus we have : $$P\left(\frac{u_{\frac{\alpha}{2}}}{\bar{X}}\leq\lambda\leq\frac{u_{1-\frac{\alpha}{2}}}{\bar{X}}\right)=1-\alpha$$ where $u_{\frac{\alpha}{2}}$ and $u_{1-\frac{\alpha}{2}}$ are the quantiles of $\Gamma(n,n)$ with levels $\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$.