I have been confused by two separate questions (Stock & Watson - introduction to econometrics ch.3), using different values for standard errors.
The first: In a survey of 400 voters, 215 respond to vote for the incumbent, 185 for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey and $\hat p$ be the fraction of survey respondents that prefer the incumbent.
Now for the variance it is given by $\hat p(1-\hat p)/n$ and when calculating the $SE(\hat p)$ we have to take the square root of the variance to get $0.0249$, and I am fine with this.
The second question: In a given population 11% of voters are African American. A survey using a random sample of 600 landline telephone numbers finds 8% African Americans. Is there evidence the survey is biased?
Now when calculating the t-statistic we use the null hypothesis with $p=0.11$, but it then states that $se(\hat p)=\hat p(1-\hat p)/n$.
Why do we no longer have to take the square root of the value above to find the standard error? I imagine it must be to do with knowing the population variance?