Exercise 8. Standard error of d

Our estimate for the difference in proportions of Democrats and Republicans is $d = \bar{X} - (1-\bar{X})$.

Which derivation correctly uses the rules we learned about sums of random variables and scaled random variables to derive the standard error of $d$?

Possible answers

$$ \begin{eqnarray} \mbox{SE}[\bar{X} - (1-\bar{X})] &=& \mbox{SE}[2\bar{X} - 1] \ &=& 2\mbox{SE}[\bar{X}] \ &=& 2\sqrt{p/N} \end{eqnarray} $$

$$ \begin{eqnarray} \mbox{SE}[\bar{X} - (1-\bar{X})] &=& \mbox{SE}[2\bar{X} - 1] \ &=& 2\mbox{SE}[\bar{X} - 1] \ &=& 2\sqrt{p(1-p)/N} - 1 \end{eqnarray} $$

$$ \begin{eqnarray} \mbox{SE}[\bar{X} - (1-\bar{X})] &=& \mbox{SE}[2\bar{X} - 1] \ &=& 2\mbox{SE}[\bar{X}] \ &=& 2\sqrt{p(1-p)/N} \end{eqnarray} $$

$$ \begin{eqnarray} \mbox{SE}[\bar{X} - (1-\bar{X})] &=& \mbox{SE}[\bar{X} - 1] \ &=& \mbox{SE}[\bar{X}] \ &=& \sqrt{p(1-p)/N} \end{eqnarray} $$