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In hypothesis testing, the common approach is to first set a null hypothesis and a hypothesis we want to test. Then apply some statistical techniques and see whether the observation is likely to happen under the assumption If null hypothesis is true. If the likelihood is low we will then reject the null hypothesis and claim our assumption is true.

But why we are not doing this the other way round? That is setting only the assumption we want to test and look at the observation. By applying some statistical technique I believe we can get how likely the observation happens under the assumption/hypothesis. We can then just use this probability to accept/reject hypothesis.

Why we are not using the second approach? It seems indirect to me in the first approach.

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The second approach is also used. For example, you want to test, if your sample is from gamma distribution family. This would be your null hypothesis $H_0,$ and if you don't set alternative hypothesis $H_1$ (such as "sample is from beta distribution family"), it would be automatically set as "sample is not from gamma distribution family". The problem is if you choose such a general alternative, the power of your test will decrease (probability of type II error will increase).

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Yes, you are asking a lucid question. How does it make sense to assess the data while assuming the null hypothesis is true? Shouldn't we rather use the data to evaluate the hypotheses? Yes, the latter makes much more intuitive sense.

The former idea --- what we often do --- is null hypothesis significance testing (NHST). You can read up on the history of NHST, and in what ways it is controversial.

As to the question, I'm not qualified to answer, so you can consider this some idle conjecture. But I think some of the reasons why we use NHST is that some of the tests are well known, have been in use for a relatively long time, and are relatively easy to conduct. I also think that if you understand what the results mean, that they are useful. In my opinion, a lot of the criticism of NHST comes down to the fact that people often misunderstand the results of the hypothesis tests.

You might look up Bayesian analysis, which could be thought of as a way to assess hypotheses based on the observed data. This makes intuitive sense, but is less straightforward to conduct than, say, something like a t-test.

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