# confidence interval for the variance with unknown distribution

Let $$X_i \sim \text{iid}(\mu, \sigma^2)$$ (the question does not specify whether or not $$\mu$$ and $$\sigma^2$$ are known). I have to show the confidence interval for the variance. Since $$X_i$$ is not necessarily normal, I should use the asymptotic distribution. Let $$\hat{\theta}^2_n = \frac1n\sum_{i=1}^n(X_i - \bar{X})^2$$. Then, $$\frac{\sqrt{n}(\hat{\theta}^2_n-\sigma^2)}{\sigma_{\hat{\theta}}}\xrightarrow{d} N(0,1).$$ Then, the asymptotic confidence interval at 5% critical level should be $$P(\hat{\theta}^2_n - \frac{1.96\sigma_{\hat{\theta}}}{n} \le \sigma^2 \le \hat{\theta}^2_n + \frac{1.96\sigma_{\hat{\theta}}}{n}) = 0.95.$$ Am I on the right track? what should $$\sigma_{\hat{\theta}}$$ be in this case?