I know the two-tailed test procedure, that's not the issue. I am wondering about the philosophy about it.

The null hypothesis $ \mu = X $ makes sense to me, we assume a certain value for the population parameter. Frequentist interpretation, the population parameter is unknown but fixed.

The alternative hypothesis $ \mu \ne X $ doesn't. If the population parameter is only slightly off from what we assume, then it is true. [Edited] If we interpret that statement probabilistically (as in Bayesian statistics), then it is almost sure it is true. There is just no point testing.

It only makes sense to me if the null hypothesis is $ \mu $ close to $ X $, and therefore the alternative hypothesis becomes $ \mu $ is not close to $ X $.

Am I making sense? Or am I mixing apples with oranges when I use frequentist/Bayesian interpretation the understand hypothesis testing?

Editing notes, this is what I wrote the [edited], thank Dave for correcting my mistake.

If we use the frequentist interpretation, then it is weird because there is a range of value such that $ \mu = X $ is true, but then $ \mu $ assume multiple values.

That statement is simply not true, see comments for what I learnt.

  • 2
    $\begingroup$ There isn’t a range of values of $X$ that makes $\mu=X$ a true statement. Either the two numbers are bang-on equal, or they are not equal. $\endgroup$ – Dave Apr 26 '20 at 7:26
  • $\begingroup$ Dave - that is my view too. That's what frequentist would say. However, in the procedure, we can establish the truth of the null hypothesis for a range of X (namely, the one without the confidence interval), that's what I think the paradox is. $\endgroup$ – Andrew Au Apr 26 '20 at 21:55
  • $\begingroup$ What do you mean by “establish the truth of the null hypothesis for a range of X” and “the one without the confidence interval”? $\endgroup$ – Dave Apr 26 '20 at 21:59
  • $\begingroup$ That was a typo, when I wrote "without", I really meant "within". I guess I know what is wrong with me now. When I cannot reject the null hypothesis, the test is inconclusive, it doesn't mean the null hypothesis is right. $\endgroup$ – Andrew Au Apr 26 '20 at 22:04

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