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My question is about hypothesis testing. In hypothesis testing we want to put the hypothesis we want to prove in the alternative hypothesis since we fix the typeI error but cannot make any statements about the typeII error. Hence, we can't draw any conclusion if we reject the null. But why is the null hypothesis in two-sided Gauß test $H_0: \mu = \mu_0$? Similarly, for all tests for normality, the nullhypothesis is $H_0: "sample comes from normal distribution". But why should we want to prove that some expectation is not a single number or does not have a specific distribution, that's not much helpful. I think we always test in the wrong direction.

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    $\begingroup$ What do you propose we test instead? $\endgroup$
    – Dave
    Jul 12 '21 at 18:42
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    $\begingroup$ One can certainly draw a conclusion went the null is rejected. The conclusion is that the null is inconsistent with the data, because the probability of observing the data we did is very small under the null. There is also one-sided hypothesis testing, where the null is a composite, like $\delta \le 0$ or $\delta \ge 0$. Normality can be important for some types of statistical tests, so that is why we see a lot of these tests taught. You can also test that something has a Poisson distribution and many others. $\endgroup$
    – dimitriy
    Jul 12 '21 at 19:04
  • $\begingroup$ @DimitriyV.Masterov But if the test rejects the null, we only know that the data is NOT normally distributed but we don't know the real distribution. Why don't we put hypothesis of normal distribution in the alternative hypothesis? Then a rejection of the null would say that data is normal distributed. Same in the two-sided tests why don't we say that $\mu=\mu_0$ is the alternative distribution, so we have evidence that $\mu_0$ is the expectation if the null is rejected? $\endgroup$ Jul 12 '21 at 21:27
  • $\begingroup$ You're basically talking about interval estimation. See this question: stats.stackexchange.com/questions/179902/… $\endgroup$
    – rishi-k
    Jul 12 '21 at 21:30
  • $\begingroup$ If we reject null the data has some particular distribution, we can't use the test that relies on that distributional assumption. This is useful info. In what you propose, there is an infinite number of distributions, so you cannot stick all of them in the null because we would have to calculate a p-value under each one. So we can never know the true distribution. Ditto for two sided tests. $\endgroup$
    – dimitriy
    Jul 12 '21 at 21:59
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We define the null hypothesis with equality because we need some null value to compute the test statistic and p-value. Consider the steps in computing the test statistic in a $t$-test.

$$ t = \dfrac{\bar x - \mu_0}{s/\sqrt{n}} $$

In order to calculate this, we need some $\mu_0$ that we define in the null hypothesis, $H_0: \mu = \mu_0$.

If we define the null hypothesis as $H_0: \mu\ne\mu_0$, we cannot calculate the test statistic.

As you have noticed, this may be of dubious utility, and alternatives have been proposed that range from confidence intervals to equivalence testing (such as TOST) to Bayesian methods.

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