This is a HW problem in my undergrad Statistics course. I am unsure of how to proceed. We were given a text file of data. I assumed that we would need $\overline{X}$; which I found to be $1.312670963$. In a past assignment we showed that as
\begin{equation}
n\rightarrow \infty, \tilde{\theta} = \sqrt{\dfrac{1}{2n}\sum_{i=1}^nX_i^2}\xrightarrow{d}N\bigg(\theta,\dfrac{\theta^2}{4n}\bigg)
\end{equation}
We are asked to construct an approximate $95\%$ confidence interval for $\theta$. If I have the Fisher function as $J(\theta)=\dfrac{4n}{\theta^2}$. Can I state
\begin{equation}
\bigg(\hat{\theta}-\dfrac{1.96\hat{\theta}}{2\sqrt{n}},\hat{\theta}-\dfrac{1.96\hat{\theta}}{2\sqrt{n}}\bigg)
\end{equation}
Using the score function and setting it to $0$. I found:
\begin{equation}
\hat{\theta}=\sqrt{\dfrac{1}{2n}\sum_{i=1}^nX_i^2}
\end{equation}
Would I be correct in using this in the above equation? I am not sure how to start this problem. Any help would be appreciated. Thanks