I know that a test statistic is used to help us in hypothesis testing, etc. We compute the test statistic, and then compare it to the $\alpha$ value to reject or accept the null hypothesis.
For a normal distribution, this is easy, you just do $ Z = ((X-\mu)\sqrt n)/\sigma $, and all of these are well-defined.
So, if I'd like to find the test statistic of a binomial distribution, I thought that it was a similar process, given the Central Limit Theorem:
$ \mu = np \\ \sigma = \sqrt(np(1-p)) \\ Z = ((X-np)\sqrt n)/\sqrt(np(1-p)) $
So, why when I look up the test statistic, I see sources such as this one under the section "Normal Approximation to the Binomial Distribution", which define the test statistic differently for a binomial distribution?
What is the test statistic used for a binomial distribution?