What is wrong with one tailed z-tests for a proportion? After answering How to analyze observed vs expected when expected is just a proportion? by suggesting that the OP uses a one tail Z-test for their proportion data, I got into a debate in the comments with another user to which their point is lost on me.
In my answer, I advocated testing $H_0: p=1/2 vs. H_1: p<1/2$. I advocated a left tail test because the OP's observed proportion was 0.2.
In the comments the other user and I said:

Alexis: Your alternate hypothesis does not correspond to the compliment of the null.
Statman:  I'm advocating a one tail test.
Alexis: Then pose the proper one-tail null hypothesis: $H_0:p\ge1/2$.
Statman: But why @Alexis? The observed p is 0.2<0.5 

I didn't receive an answer, but received a healthy downvote.
Now the left tail test is given as a valid option in textbooks, e.g. in Introduction to Probability and Statistics by Milton et. al, so my question is what point is @Alexis trying to make about left tailed z-tests?
 A: My comment was specifically about your articulation of the appropriate one-sample one-sided (aka one-tailed) null hypothesis (not about one-sided tests per se) which, for proportions, should either be $H_{0}: p \ge p_{0}$ with $H_{1}: p < p_{0}$, or $H_{0}: p \le p_{0}$ with $H_{1}: p > p_{0}$. Bear in mind that null hypotheses are articulated before you evaluate your data for directionality of a rejection decision.
The null hypothesis you posed in your answer was of the form $H_{0}: p = p_{0}$ which has as its proper alternative $H_{1}: p \ne p_{0}$, since by definition an alternative hypothesis corresponds to the complementary event in the null hypothesis. However, you proposed an alternative $H_{1}: p < p_{0}$, which is not the complement of your null. Indeed, it does not even correspond to the alternative hypothesis expressed on the site you linked to in your answer. The crux of the issue is that it may truly be the case that $p>p_{0}$, but this state of nature does not fit within either your $H_{0}$ or your $H_{1}$, since the sample space of $p$ is not fully represented by your null and alternate, they cannot be well formed.
To be super explicit: Nothing is wrong with one-sided one-sample inequality tests, but you articulated the null hypothesis incorrectly for such a test.
A: One problem with your suggestion is that the normal approximation for proportions is only reasonable in certain circumstances. A large sample and the observed proportion well away from the boundaries of 0 and 1. Neither of those circumstances is specified in the original question. See this question: Testing equality of two binomial proportions proportion (one near 100 %)
I have previously sparked some discussion by suggesting that the normal approximation method for confidence intervals of proportions be omitted from textbooks. You can read it here: What statistical methods are archaic and should be omitted from textbooks?
