Estimating target variable without data I have a metric labor hrs per unit, $L$, that is currently tracked by:$$\frac{\text{clocked worker hours}}{\text{total output}} $$
on a per week basis. Unfortunately, it does not matter what type of unit is being worked on, the labor hrs for a specific type of unit is not accounted for in the denominator.  There is a particular unit denoted by $U$, that I need to estimate $L$ for. Hence, I think I might want to estimate something like: $$L_U = \frac{\text{clocked worker hours}}{\text{total output for $U$}}$$
There is no data available for something like this and all I have is the amount $L$  per week and the count of $U$ that comes in every week (the incoming $U$ does not mean that all of it is going to be part of the total output for $U$, some of those incoming units get scrapped and do not count in the overall output).
The only thing I can think of is to collect live samples from a time study and use Kernel Density Estimation (KDE) to estimate the distribution of $L_U$ (after I have figured out what features may effect this from the domain experts). Once the distribution is estimated, I can randomly sample "synthetic" data from it and use it to estimate $L_U$. I am also going to try to do some rule-based modeling based on the estimates/intuition of domain experts but I feel like that might be a long shot.
Question 1:
Would like to know how others would tackle this problem (preferably without a time study/collecting live samples) and if my current approach makes sense to even attempt?
Question 2:
If I were to be able to estimate this with the sheer lack of data at my disposal, how would I provide a measure of confidence/certainty? Surely, there would be a massive lack of statistical integrity?

EDIT: I was able to get access to real-life samples, but $< 10$. I have the current framework set up for this project of mine, and other particular questions:
Assume you have $< 10$ real-life observations of a variable X.

*

*Step 1: Use Kernel Density Estimation on the $< 10$ samples to
approximate the distribution for $X$.


*Step 2: Randomly sample 10,000 observations from the resulting
distribution from Step 1 therefore creating synthetic data.


*Step 3: Iterate on Step 2 using MC simulation to extract estimates
for the population mean and variance of $X$.
Questions:

*

*Question 1: Does this approach seem robust? Should I verify it with a hypothesis test by gathering a decent number of real samples once I am able to? What other measures of confidence can I use?


*Question 2: Should the KDE step be incorporated into the MC
simulation? If so, how/why?


*Question 3: Does my choice of kernel matter? If so, how do I choose the appropriate type of kernel?

< 10 is probably not enough samples for KDE. See the answer to a specific question about KDE and sample size  (reference given in the answer).
If you have $U$ and worker hours (the numerator of $L$), you can actually use simple linear regression to infer worker hour per unit of $U$ produced.
Let clocked worker hours =$C$
$C= \beta_0 + \beta_1*U +e$
The estimate for $\beta_1$ is the change in worker hours for a one-unit increase in $U$. This assumes a linear relationship, that there are no confounders (i.e. there's no underlying process like "When we increase production of $U$, we always increase this other labor-intensive product $V$."), and that your values are independent.
You can use your actual data as a hold out sample to see if it matches the relationship you've drawn. In my opinion, that small sample of real data is most valuable as a test set for estimates you are getting with the larger data.
