Another Question from Gelman - Regression & Other Stories The question:
Compare two options for a national opinion survey: (a) a simple random sample of 1000 Americans, or (b) a survey that oversamples Latinos, with 300 randomly sampled Latinos and 700 others randomly sampled from the non-Latino population. One of these options will give more accurate comparisons between Latinos and others; the other will give more accurate estimates for the total population average.
(a) Which option gives more accurate comparisons and which option gives more accurate population estimates?
(b) Explain your answer above by computing standard errors for the Latino/other comparison and the national average under each design. Assume that the national population is 15% Latino, that the items of interest are yes/no questions with approximately equal proportions of each response, and (unrealistically) that the surveys have no problems with nonresponse.
My main problem is why part b) of this question. It seems fairly obvious that the answer to part a) is the oversampled Latino design will provide a better estimate for comparison between Latinos and others, because the sample will twice the proportion of Latinos found in the population (15%), while the random sample of 1000 is more representative of the population.
But I'm really at a loss where to even start with part b). I have no idea how I am to model responses from a survey without being given any estimates from the survey results.  If someone would be willing to offer some pointers/ a walkthrough I would be highly grateful.
 A: Since the problem is a yes/no answer, we can model the outcome as binary (1-- yes, 0-- no).
The standard error of the outcome is the estimate of the standard deviation divided by the square root of the sample size. Hence, the standard error is
$$ \dfrac{\sigma}{\sqrt{n}} \>.$$
All we need do is estimate the standard deviation.  Let's assume our main interest is the proportion of yes's rather than the raw count.  Our outcome can then be considered a Bernoulli random variable for which the standard deviation is
$$ \sigma = \sqrt{p(1-p)} \>. $$
Here, $P(y=1)=p$.  The question notes that the answers are roughly the same, so let's assume $P(y=1)=P(y=0)=0.5=p$.  The estimate of the standard error for the Latino response in the simple random sample is
$$ \dfrac{\sigma}{\sqrt{n}} = \sqrt{\dfrac{0.15 \times 0.85}{150}}\>.$$
We use $n=150$ because the question states the national population is 15% Latino, so 15% or 150 of our 1000 samples should be Latino.
The case for the oversampling of Latinos is similar, with standard error
$$ \dfrac{\sigma}{\sqrt{n}} = \sqrt{\dfrac{0.15 \times 0.85}{300}} \>.$$
Now we understand why the oversampling leads to a better comparison; we achieve more precision in the Latino response by oversampling.
