Misunderstandings caused by the standard paradigm of

hypothesis tests

A frequentistic hypothesis test follows the
scheme outlined below (see Fig. ).
^{1.10}

- Formulate a hypothesis .
- Choose a test variable of which the probability density function is known (analytically or numerically) for a given .
- Choose an interval
such that there is
high probability that falls inside the interval:
(1.6)

with typically equal to 1% or 5%. - Perform an experiment, obtaining .
- Draw the following conclusions :
- if accepted;
- otherwise rejected with a significance level .

The usual justification for the procedure is that the probability is so low that it is practically impossible for the test variable to fall outside the interval. Then, if this event happens, we have good reason to reject the hypothesis.

One can recognize behind this reasoning a revised version of the classical `proof by contradiction' (see, e.g., Ref. [10]). In standard dialectics, one assumes a hypothesis to be true and looks for a logical consequence which is manifestly false in order to reject the hypothesis. The slight difference is that in the hypothesis test scheme, the false consequence is replaced by an improbable one. The argument may look convincing, but it has no grounds. In order to analyse the problem well, we need to review the logic of uncertainty. For the moment a few examples are enough to indicate that there is something troublesome behind the procedure.

**Example 4:**- Choosing the rejection region in the middle
of the distribution.

Imagine choosing an interval around the expected value of (or around the mode) such that(1.7)

with small (see Fig. ). We can then reverse the test, and reject the hypothesis if the measured is inside the interval. This strategy is clearly unacceptable, indicating that the rejection decision cannot be based on the argument of practically impossible observations (smallness of ).One may object that the reason is not only the small probability of the rejection region, but also its distance from the expected value. Figure is an example against this objection.

Although the situation is not as extreme as that depicted in Fig. , one would need a certain amount of courage to say that the is rejected if the test variable falls by chance in `the bad region'.

**Example 5:**- Has the student made a mistake?

A teacher gives to each student an individual sample of 300 random numbers, uniformly distributed between 0 and 1. The students are asked to calculate the arithmetic average. The prevision^{1.11}of the teacher can be quantified with

E (1.8) (1.9)

with the random variable normally distributed because of the central limit theorem. This means that there is 99% probability that an average will come out in the interval :(1.10)

Imagine that a student obtains an average outside the above interval (e.g. . The teacher may be interested in the probability that the student has made a mistake (for example, he has to decide if it is worthwhile checking the calculation in detail). Applying the standard methods one draws the conclusion that*``the hypothesis = `no mistakes' is rejected at the 1% level of significance'',**``there is only a 1% probability that the average falls outside the selected interval, if the calculations were done correctly''.*^{1.12}i.e. that concerning the probability of mistake, and not that of results far from the average if there were no mistakes. Moreover, the statement sounds as if one would be 99% sure that the student has made a mistake! This conclusion is highly misleading.How is it possible, then, to answer the very question concerning the probability of mistakes? If you ask the students (before they take a standard course in hypothesis tests) you will hear the right answer, and it contains a crucial ingredient extraneous to the logic of hypothesis tests:

*``It all depends on who has made the calculation!''* **Example 6:**- A bad joke to a journal.
^{1.13}

A scientific journal changes its publication policy. The editors announce that results with a significance level of 5% will no longer be accepted. Only those with a level of will be published. The rationale for the change, explained in an editorial, looks reasonable and it can be shared without hesitation:

*``We want to publish only good results.''*1000 experimental physicists, not convinced by this severe rule, conspire against the journal. Each of them formulates a wrong physics hypothesis and performs an experiment to test it according to the accepted/rejected scheme.

Roughly 10 physicists get significant results. Their papers are accepted and published. It follows that, contrary to the wishes of the editors, the first issue of the journal under the new policy contains only wrong results!

The solution to the kind of paradox raised by this example seems clear: The physicists knew with certainty that the hypotheses were wrong. So the example looks like an odd case with no practical importance. But in real life who knows in advance with certainty if a hypothesis is true or false?