Re-sampling at defined percentage of the population Background: I have data that contains the severity scores of the pathology findings of tissues from different subjects with 100% sampling of the organ (let's call this dataset population, which contains thousands of scored slides for each subject). Certainly, this type of scoring will generate great data however, to reduce the resources needed, pathologists typically sample a fraction of the organ.
Objective: My goal is to find out the least acceptable fraction sampling that gives us enough information to assess the disease.
Method: I am going to use resampling (with replacement) of my data (population) at different fractions (5%, 10%, etc.) and calculate the coefficient of variation (CV), CE to summarize the results and provide it as a guideline for researcher to pick their accuracy of interest (based on CE, CV, SD, etc) and perform their sampling.
Additional information: Two different pathologists scored the fields in each slide. They used a semi-quantitative (0 to 4; with 1 increment) scoring system. We will use the average scores of two readings per field of view in every slide (so our data will be at the field of view level). There are up to 52 slides per organ, and slides have up to 42 fields of view.
Questions:
(i) What is the best way to show sampling at a certain fraction is enough? In other words, is this correct to use a test (e.g., t-test) to compare the mean of different fraction sampling with the true mean (mean of the population) and provide a p-value and call it enough fraction sampling if the test does not find a sig. difference?
(ii) Bootstrapping uses the same number of the dataset; however, here we are doing resampling at different fractions of the population. What type of sampling is this?
 A: As Tim said in a comment, what you're doing is re-sampling, not a bootstrap. There is a type of bootstrap, often called the"$m$ of $n$" bootstrap," in which your resample sizes $m$ aren't of the same size as your original data sample $n$. That can be useful for estimates of difficult statistics like extremes; see this page and its links. But you are evaluating something more like a mean value, and you specifically want to evaluate how your ability to estimate it changes with the sample size.
One problem if you "provide a p-value and call it enough fraction sampling if the test does not find a sig. difference" is that you can find a non-significant difference just by having a really small sample size. What you need is some type of equivalence test.
One approach is to do two one-sided tests (TOST). You decide first on lower and upper limits around  your true value that don't matter in practice. You then perform one-sided tests of two null hypotheses on the re-sampled values: one that that the mean re-sampled value is below than the lower limit and one that it is above the upper limit. If you reject both those null hypotheses then you accept equivalence. That forces you to have enough data to reject those properly formulated null hypotheses.
Before jumping to that, however, look carefully at the results of re-sampling. I'd recommend looking carefully at histograms of the 5 score levels (without averaging yet) over multiple re-samples at each of several sampling percentages.
I'd watch out for problems that might come from intra-organ heterogeneity. You might have the ability to sample randomly from all portions of the organ, but will that happen in practice? If you find that 5% sampling is adequate, will the pathologist still prepare the whole organ and then randomly choose 5% as you did in your evaluation? Or will only 5% of the organ be prepared and evaluated? Unless you can count on great homogeneity within each organ that's evaluated, those won't be the same. See for example: Kayser, K., Schultz, H., Goldmann, T. et al. Theory of sampling and its application in tissue based diagnosis. Diagn Pathol 4, 6 (2009).
