The Statement of the Problem:
Suppose that $Y_1, Y_2, \ldots, Y_n$ constitute a random sample of size $n$ from an exponential distribution with mean $\theta$. Find a $100(1-\alpha)\%$ confidence interval for $\theta$.
Where I Am:
I suppose that what I am to do here is construct a CI of the form
$$ \hat \theta \pm z_{\frac{\alpha}{2}} \frac{1}{\sqrt{nI(\hat \theta_\text{mle})}}.$$
Well, I found $\hat \theta_\text{mle}$ (where $\theta = 1/\lambda \implies \lambda = 1/\theta)$:
$$ \hat \theta_\text{mle} = n\sum_{i=1}^n Y_i.$$
But how do I find the Fisher Information of this MLE?
I have in my notes that
$$ \hat \theta_\text{mle} \approx N\left(\theta_0, \frac{1}{nI(\theta_0)}\right) $$
which I assume I'm supposed to use... but I'm not sure what this is saying. Do I have to find the Fisher information of a normal distribution with these parameters? That... doesn't make sense to me. If anyone could shed some light on this, I'd appreciate it. Thanks.