Suppose we want to verify the probability a sample comes from a distribution with mean $\mu_1$ and variance $\sigma²_1$. Then, we conduct a hypothesis test and decide to reject the null hypothesis, i.e., we consider that is improbable the sample was drawn from a distribution with these parameters.
Now, we want to estimate $\mu_2$ (the population mean from which the sample was drawn) using Confidence Interval. Should we use $\sigma²_1$ or $s²$ (sample variance) when calculating the CI? I think, since we rejected the hypothesis null, there's no reason to consider any parameter from the null distribution. We have decided that our sample doesn't come from that distribution, so we should only use estimated parameters to determine the confidence interval. Is this conclusion correct?
It's not the second moment