For a z-test for one proportion, a possible effect size measure is (p-p_0)/(sqrt(p)-(1-p_0))) where P_0 is the null value and p is the true population proportion, which can be estimated using the sample proportion. What is the relationship between this effect size and the z-test statistic for this situation? Does the effect size fit the relationship "Test statistic = Size of effect * Size of study? If not, explain why not If so, show how it fits. What would be a reasonable way to estimate this effect size?

a) Remember that the formula for the $z$-statistic for one-proportion is,
$z=\frac{\text{Statistic}-<!--\; -\; -->\text{Null value}}{\text{Null standard error}}=\frac{\stackrel{^<!--\; ^\; -->}{p}-<!--\; -\; -->p_0}{\sqrt{\frac{p_0\cdot <!--\; \cdot \; -->(1-<!--\; -\; -->p_0)}{n}}}=\frac{\stackrel{^<!--\; ^\; -->}{p}-<!--\; -\; -->p_0}{\sqrt{p_0\cdot <!--\; \cdot \; -->(1-<!--\; -\; -->p_0)}}\cdot <!--\; \cdot \; -->\sqrt{n}$,where $\stackrel{^<!--\; ^\; -->}{p}$​ is the observed value of the sample proportion, $p_0$​ is the null value, $n$ is the sample size and $d=\frac{\stackrel{^<!--\; ^\; -->}{p}-<!--\; -\; -->p_0}{\sqrt{p_0\cdot <!--\; \cdot \; -->(1-<!--\; -\; -->p_0)}}$ ​​is the (estimated) effect size. So, this effect size is actually,$\boxed{{\displaystyle d=\frac{z}{\sqrt{n}}}}$, where $z$ is the value of test statistic and $n$ is the sample size.
b) We've already determined in a) part that $z=d\cdot <!--\; \cdot \; -->\sqrt{n}$, which in fact means that the value of test statistic is the effect size multiplied by the "size of the study" (i.e. square root of the sample size), so the effect size does fit the mentioned relationship.
c) The most reasonable way to estimate this effect size would be to use the value of the sample proportion $\stackrel{^<!--\; ^\; -->}{p}$​ instead of the population proportion $p$ (because we don't know the population proportion) in the formula. Additionally, population proportion $p$ is the only parameter in the formula for the effect size that we don't know, so that's the only value that we should approximate. The sample proportion is clearly the most suitable choice. Notice that we've used this approximation to answer parts a) and b), since using the population proportion ppp renders the questions in a) and b) part meaningless (how could there be any relationship between effect size and test statistic, if test statistic uses sample proportion, while effect size uses population proportion?).
Answer: a) The relationship is $d=\frac{z}{\sqrt{n}}$; b) Yes; c) Use sample proportion $\stackrel{^<!--\; ^\; -->}{p}$​ instead of the population proportion $p$.