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I recently read the following regarding the null and alternative hypotheses:

If the original claim includes equality $\left(\le, =, \text{or} \ge\right)$, it is the null hypothesis. If the original claim does not include equality $\left(<, \ne, >\right)$ then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign.

I was until now under the impression that our original claim was always the alternative hypothesis. I thought it was insufficient to merely fail to reject our claim – we need to actively have enough evidence to reject the null and accept our claim.

I do agree that have a non-closed set (say, $\mu < \mu_0$) as our null can occasionally be misleading, and is in most cases equivalent to have a closed set ($\mu \le \mu_0$). But, does this really mean we should define the null & alternative hypothesis solely based on the equality sign?

Guillaume F.
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    Adding the source for the exact claim: these course notes. – Arya McCarthy Apr 12 '21 at 22:43
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    I would abandon those notes because they are incorrect and misleading. – whuber Apr 13 '21 at 13:33
  • @whuber could you elaborate on what's wrong with the notes? – Guillaume F. Apr 13 '21 at 15:24
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    Each of the five statements in the description of the null hypothesis is misleading and wrong. They mislead by suggesting an examination of inequalities is a useful or correct way to identify a null hypothesis. They are wrong by supplying invalid recommendations: there are common, important exceptions to everything recommended in those notes. They are also ambiguous by not explaining or defining nonstandard (but crucial terms) such as "original claim." – whuber Apr 13 '21 at 16:39
  • It is possible to make a claim based on the null hypothesis, but they would be based on the statistcal power of the test. ISTR there is a statistical term for that sort of testing, but I can't recall what it is. – Dikran Marsupial Sep 03 '22 at 10:10
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    @DikranMarsupial Equivalence testing? – Dave Sep 03 '22 at 10:14
  • @Dave, that isn't the one I had in mind, but certainly seems relevant to the question. The example I had in mind was that climate skeptics often point out the lack of statistical warming since [cherrypicked start date] as evidence against climate change. However the period is invariably too short for a failure to reject the null to be meaningful. If they had a period long enough to have high statistical power, their argument would have some value. So this isn't arguing for/against equivalence, but explicitly for H0 as the aim of the test. – Dikran Marsupial Sep 03 '22 at 10:42
  • So in that case, you can construct an argument where you are arguing for the null, but evaluating the power of the test is often quite tricky, which is why it is normally much better to have a "devils advocate" null hypothesis, which opposes the thing you are arguing for. Unfortunately a lot of people adopt the "null ritual" and think that H0 has to be an assumption of no difference. – Dikran Marsupial Sep 03 '22 at 10:45
  • From the notes mentioned above by @AryaMcCarthy: "Null Hypothesis ( H0 ) Statement of zero or no change." oh dear... https://www.sciencedirect.com/science/article/pii/S1053535704000927 Fullly agree with whuber! – Dikran Marsupial Sep 03 '22 at 16:11

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