I feel like OP unfortunately mostly misses the point of Gosset's work and the t-test, in the usual way of people taught the contemporary bastardization of Gosset/Fisher/Pearson/Neyman as NHST. The important thing isn't that it lets you calculate some _p_-value; the important thing is that it is a framework for decision-making, where you can trade off your false-positive and your false-negative rates to make the economically rational decision: https://www.tandfonline.com/doi/full/10.1080/00031305.2018.1...
In this case, because it is a well-understood problem with the rates of bad batches easily established from the brewery's records of testing & drinking, it lets the brewery decide on how many 'off' batches it wants to risk in exchange for saving the cost of a certain number of test-samples. You decide you want to risk 1 bad batch in 100 for a false negative while rejecting 1 good batch in 20, then you need _n_ samples etc. And this directly translates better measurements (by lowering variance, eg. by blocking) into money: the lower the variance, the fewer samples you need to achieve any given tradeoff, thereby saving the brewery money on scrapped material or testing. The smaller the better, hence Student's inability to use asymptotics or approximations: they might be off by orders of magnitude. (He would even try to do _n_ = 2 tests!)
Or they might be trying to tightly optimize alcohol content, to avoid taxation for passing high-alcohol content thresholds, but also avoid going too low to disappoint their customers, so Student would explicitly calculate out scenarios, for example:
> Thus, Gosset concluded, βIn order to get the accuracy we require [that is, 10 to 1 odds with 0.5 accuracy], we must, therefore, take the mean of [at least] four determinations.β The Guinness Board cheered. The Apprentice Brewer found an economical way to assess the behavior of population parameters, using very small samples.
(If you're thinking this sounds like a very subjective-Bayesian decision-theory thing to write, you are right, although Student would have rejected that, like most statisticians, and emphasized that he was dealing with populations with known base rates, and so nothing Bayesian was necessary; it was just a frequentist decision-theory approach.)
In this case, because it is a well-understood problem with the rates of bad batches easily established from the brewery's records of testing & drinking, it lets the brewery decide on how many 'off' batches it wants to risk in exchange for saving the cost of a certain number of test-samples. You decide you want to risk 1 bad batch in 100 for a false negative while rejecting 1 good batch in 20, then you need _n_ samples etc. And this directly translates better measurements (by lowering variance, eg. by blocking) into money: the lower the variance, the fewer samples you need to achieve any given tradeoff, thereby saving the brewery money on scrapped material or testing. The smaller the better, hence Student's inability to use asymptotics or approximations: they might be off by orders of magnitude. (He would even try to do _n_ = 2 tests!)
Or they might be trying to tightly optimize alcohol content, to avoid taxation for passing high-alcohol content thresholds, but also avoid going too low to disappoint their customers, so Student would explicitly calculate out scenarios, for example:
> Thus, Gosset concluded, βIn order to get the accuracy we require [that is, 10 to 1 odds with 0.5 accuracy], we must, therefore, take the mean of [at least] four determinations.β The Guinness Board cheered. The Apprentice Brewer found an economical way to assess the behavior of population parameters, using very small samples.
(If you're thinking this sounds like a very subjective-Bayesian decision-theory thing to write, you are right, although Student would have rejected that, like most statisticians, and emphasized that he was dealing with populations with known base rates, and so nothing Bayesian was necessary; it was just a frequentist decision-theory approach.)