Problem: Let $X$ be $Bin(n,p)$ and $\hat p = X/n$, and $$Z=\frac{\hat p - p}{\sqrt{p(1-p)/n}}$$ We see that $Z$ is asymptotically $N(0,1)$. Find a $1-\alpha$ confidence interval for $p$. Note that this is very different from the regular $$\frac{\hat p - p}{\sqrt{\hat p(1-\hat p)/n}}$$
Attempt: I think I should use a Wald (or perhaps a Score) interval to solve this. I have found this document that describes how it's done, but I don't know if it's the right approach. Furthermore, it's an undergrad course in probability so we don't know very much.
As @BruceET pointed out this a 'Wilson binomial confidence interval' see https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
We can use the formula $$\frac{1}{1 + \frac{1}{n} z_{\alpha/2}^2} \left[ \hat p + \frac{1}{2n} z_{\alpha/2}^2 \pm z_{\alpha/2} \sqrt{ \frac{1}{n}\hat p \left(1 - \hat p\right) + \frac{1}{4n^2}z_{\alpha/2}^2 } \right]$$