Let's say that somehow $100(1-\alpha)\%$ confidence interval of population mean $\mu$ is known as $(a,b)$ and the number of samples is $n$. Is it possible to infer point estimates of population mean and population variance from this information? In this case, the assumption is that the population follows normal distribution.
One idea is that because confidence interval of population mean can be calculated if we know sample mean $\overline{x}$ and population variance $\sigma^{2}$: $$\overline{x}-z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\leq\mu\leq\overline{x}+z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$$ , we can set $a=\overline{x}-z_{\alpha/2}\frac{\sigma}{n}$, $b=\overline{x}+z_{\alpha/2}\frac{\sigma}{n}$ and solve for $\overline{x}$ and $\sigma$. Certainly, in this case, $\overline{x}$ can be treated as point estimate of population mean. However, what about $\sigma^{2}$? Is this "true" population variance or is this just "point estimate" of population variance? I am really confused about how $\sigma^{2}$ should be interpreted in this case.