Why is my method not applicable, one-sided confidence interval I have "solved" a question in mathematical statistics, but unfortunately received an incorrect answer and I would appreciate any help in understanding why my method did not work in this case. The question is as follows:

In city A, 500 randomly selected people were tested for antibodies
to covid-19, which resulted in 110 giving positive results. In city
B, a random sample of 500 people was also drawn, and here 91 had positive results.
Can it be said that it is statistically certain that city A has a
larger population share with antibodies than city B?
(Justify your answer with an appropriate test. Use 5% significance
level.)

I started by producing $P^*(A)=\frac{110}{500}=0.22$ and respectively $P^*(B)=\frac{91}{500}=0.182$.
Then I chose to continued with a one-sided hypothesis testing where I assumed that the data was binomially distributed according to $Bin(n=500,p)$, with $$H_0:P=0.182 \text{ and }H_1:>0.182.$$
And finally brought out the one-sided interval according to: $$I_p=\lambda_\alpha \sqrt{\frac{P^*(A)(1-P^*(A))}{n}}\Rightarrow [0.189,\infty].$$ And concluded that $H_0$ should be rejected since 0.182 is not included in the interval. But according to the results, $H_0$ should not be rejected and they have also used a different method than I did. Why is this method not applicable in my case?
Note: At first I thought that errors were because I only used α and not $\frac{\alpha}{2}$ in the calculation, however, even if I were to replace this, 0.182 is not in the interval.
 A: Your standard deviation formula is off by the so-called finite sample correction factor. In essence, if you sample a large piece of the total parent of size ${N}$, there is an associated reduction in expected variability (going to even zero for a sample equal size equal to ${N}$). In other words, complete sample implies complete accuracy in measuring the parent's portion of an item.
My review of the theoretical math presented in the cited source below around the selection of the required sample estimate ${m}$ (here 500) as to meeting the criteria stated remains, however, a relative function of each cities actual size (see extended discussion and formula derivation in this educational source, for example).
However, numerically assigning values from a rationale range of potential parent population city sizes may be insightful if, for example, a sample size equal to 500 (as the ${m}$) is at all possibly sufficient/insufficient to meet the required confidence level and error level specified.
