Sampling the population for Covid-19 infection rate?

Question

(1) How do we sample the population for COVID-19 to attain a 95% reliable range estimate of the infection rate for major cities?

(2) How do we select the sample (random or stratified random)?

(3) How do we determine the sample size?

(4) By what process could we generalize this to the whole population?

A sample size calculator indicated that a random sample size 1509 of the whole population of 328000000 would provide 98% confidence that the sample proportion would be ± 3% of the population proportion. We might want to stratify for, income level, race and age.

I am taking this to mean that we could randomly test 1500 people in each of the largest 50 cities (using only 75,000 tests) and get a much better understanding of the actual infection rate.

Answer 1

Regarding (1), a conservative (larger than real) standard deviation of the proportion $p$ of infected if you use random sampling could be computed as $\sqrt{0.25/n}$ where $n$ is the sample size and 0.25 an upper bound of $p(1-p)$.

Concerning (2)(3), you would have large gains to realize using stratified sampling if you have easy-to-identify groups with widely different values of $p$. This could be groups of age, urban/suburban communities, etc. Within each stratum, you can use the formula above to select the sample size.

Concerning (4), once you estimate $p$ for each stratum you "raise" that estimation weighting by the size of the stratum. So if you have, say, two strata of size $N_1$ and $N_2$ and $N=N_1+N_2$, your estimate of $p$ for the population would be $$\hat{p} = \frac{N_1}{N}\hat{p}_1 + \frac{N_2}{N}\hat{p}_2.$$

You can look any book on sampling, such as Cochran's Sampling Techniques.