Methods for drawing population inferences from multiple sub-population datasets What would be an appropriate model or method for making inferences about a broader population quantity from multiple quantities representing subsets of the population?
Imagine, as an example, that I want to estimate the total number of cars sold in the US in 2021. I have data from a number of car dealers, but not all, for new car sales in 2021. I also have data from a subset of states for used car sales in 2021. So I know how many new cars sellers a, b, and c sold. And I know how many use cars were sold in states 1, 2,…,30. I want to use these datasets to estimate the total number of cars sold in the US and put some reasonable bounds around the quantity.
What tools would one use to tackle an inference problem like this? What methods exist for combining multiple incomplete measures into a single, appropriately bounded, estimate for the full-population quantity?
 A: It depends a lot on how you acquired the data for the new and used sales. Are you sitting on a convenience sample that could be biased, or do you think it closely represents a random draw from all dealerships and states?
Convenience sample
In this case, you know close to nothing about the car sales in the US. You have to gather some data from the part of the population (dealers and states) that you don't have data, and you should do it in such a way that you have reason to believe its a random draw.
Once you've done that, you can do the inference separately on the missing part of the population (see next section "Random draws" on how to do that), add together with the data you have, and presto! The only statistical uncertainty in that estimation comes from the sampling of the missing part of the population, which you know from the estimations below.
Random draws
If the data you have are from random draws, what you have is stratified sample, where one stratum is new cars, and the other is used cars. The total number of cars is – fairly unsurprisingly – given by[1]
$$X = N_d \bar{x}_d + N_u \bar{x}_u$$
where $N_d$ is the total number of dealerships and $\bar{x}_d$ is the mean number of cars sold per dealership in your sample. Similarly, $N_u$ is the number of states and $\bar{x}_u$ is the mean number of used cars sold in the states in your sample.
The variance of this estimation is
$$\begin{split}
s^2_X &= N_d^2 \left(\frac{N_d - n_d}{N_d}\right) \frac{s_d^2}{n_d} \\ \\
 &+ N_u^2 \left(\frac{N_u - n_u}{N_u}\right) \frac{s_u^2}{n_d} 
\end{split}$$
If inter-class variance is larger than intra-class variance, this stratified estimation can have lower variance than a plain one.
[1]: Deming, 1950, Some Theory of Sampling.
But you can also find the equations for the mean on Wikipedia and scale accordingly (mean by N, variance by N²): https://en.wikipedia.org/wiki/Stratified_sampling#Mean_and_standard_error
