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To what extent does the study of history use hypothesis testing? By Hypothesis testing I mean the usage of p-values and significance levels to dictate how an actual event differs from the event by chance;

An example where it could be used in history:

To evaluate the change in attitudes of female politicians since laws in country A was changed. The sample would be all politicians of country A for a single year and evaluate the p value with assumed distribution X~B(sample size, 0.5). This can then be graphed to show that the distribution is becoming less/more/as true. The math would probably take more into account but the basic idea is the same

It seems to me, you mixed two control methods. One, the practical-level method, is statistics. Yes, it is used, alas, very often, without the understanding of the tool. Historians are seldom good at maths, and the Theory of Probability is one of its most demanding areas. It is known, that even people with good math background, but without deeper knowledge of the ToP, can be easily caught by its numerous paradoxes. (The most simple and famous one: "A family has 2 children, one of them is a boy. What is the probability that another one is a boy, too?" Answer - 1/3.) And understanding statistics without the understanding of ToP is impossible.

Second, is a much more deep and important method - predict-and-check. This is the method that divides science from non-science. It is not connected directly to statistics and, naturally, can be used and is used in history, not predicting the future, but predicting the results of excavations. They cannot be known beforehand but can be predicted by the theory you want to check. Freshly translated or newly found documents can serve the same way.

That method needn't statistics, but demands an even deeper understanding of probabilities - conditional probabilities and probabilities of conditions. Less mechanical counting, but much more thinking and understanding. (I advise HPMOR as a very good popular source and a starting point)

Thank God, the method can be often used on an intuitive level, which is practiced by 90% of scientists. Alas, that can sometimes lead to serious errors.

If you want to cross these methods and to predict the results of statistics, yes, it is possible, but you will be in the area of sociology, economics, and politology (as your example), but not of history. And seriously, you don't want it. You really must know ToP and statistics at a very good level for it. Say, 2-3 years of lections and a thousand solved practical problems. On the base of combinatorics, the theory of sets, and logic, of course.

Yes, of course. Suppose archaeologists are excavating a region. On average,they find n pottery shards per hectare. At one dig they count 5n pottery shards. They ask themselves "What are the likelihood of finding 5n pottery shards at a random dig?" Since they have estimated the parameters of the pottery shards distribution for the region they can calculate this. So they calculate it and the answer is that it is incredibly unlikely. "Aha!" they say, "here there must have been a settlement!"

This is an example of hypothesis testing. Another example is trying to reconstruct partially destroyed manuscripts. To fill in the blanks you need statistical methods to model the language the manuscript is written in.