Minimum sample size calculations for two-sample two-tailed proportion z test I'm struggling to reconcile some minimum sample size calculations. I'm seeking a minimum sample size for two-sample two-tailed proportion z test. I set my parameters as follow...

*

*alpha = 0.05

*power = 0.80

*kappa = 1

*pA = 0.45

*pB = 0.55

...and get a minimum sample size of nA = 389 and nB = 389. See here. So far so good.
But when I run a two-sample two-tailed proportion test using these resulting parameters:

*

*pA = 0.45

*pB = 0.55

*nA = 389

*nB = 389

...I get a p-value of 0.005. See here.
The problem is that I think the small p-value suggest the model is greatly over-powered, meaning I could make due with a much smaller sample size. I believe the first calculation suggests a test with these sample sizes / parameters would give me a p-value closer to alpha=0.05. In fact, I can achieve the p-value of 0.048 that I expected by using half the sample size suggested in the minimum sample size equation(nA = 195, nB = 195).
Is nB=389 and nA=389 actually the minimum sample size needed, if so why am I getting such a small p-value, and what am I doing wrong?
Thanks so much!

It's best to think of this in terms of sample size needed to achieve a certain precision (margin of error; half the width of a confidence interval) in estimating the difference in two probabilities.  As detailed here it requires $n=96$ to estimate a single probability to within a margin of error (at the 0.95 level) of $\pm 0.1$, and 196 observations to estimate the difference to within $\pm 0.1$.
