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??
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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 $1-\frac{\alpha}{2}$.
