Editing Misuse of statistics (section)

===Proof of the null hypothesis===
In a statistical test, the [[null hypothesis]] (<math>H_0</math>) is considered valid until enough data proves it wrong. Then <math>H_0</math> is rejected and the alternative hypothesis (<math>H_A</math>) is considered to be proven as correct. By chance this can happen, although <math>H_0</math> is true, with a probability denoted <math>\alpha</math> (the significance level). This can be compared to the judicial process, where the accused is considered innocent (<math>H_0</math>) until proven guilty (<math>H_A</math>) beyond reasonable doubt (<math>\alpha</math>).

But if data does not give us enough proof to reject that <math>H_0</math>, this does not automatically prove that <math>H_0</math> is correct. If, for example, a tobacco producer wishes to demonstrate that its products are safe, it can easily conduct a test with a small sample of smokers versus a small sample of non-smokers. It is unlikely that any of them will develop lung cancer (and even if they do, the difference between the groups has to be very big in order to reject <math>H_0</math>). Therefore, it is likely—even when smoking is dangerous—that our test will not reject <math>H_0</math>. If <math>H_0</math> is accepted, it does not automatically follow that smoking is proven harmless. The test has insufficient power to reject <math>H_0</math>, so the test is useless and the value of the "proof" of <math>H_0</math> is also null.

This can—using the judicial analogue above—be compared with the truly guilty defendant who is released just because the proof is not enough for a guilty verdict. This does not prove the defendant's innocence, but only that there is not proof enough for a guilty verdict.

"...the null hypothesis is never proved or established, but it is possibly disproved, in the course of experimentation.  Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis."  (Fisher in ''[[The Design of Experiments]]'')  Many reasons for confusion exist including the use of double negative logic and terminology resulting from the merger of Fisher's "significance testing" (where the null hypothesis is never accepted) with "hypothesis testing" (where some hypothesis is always accepted).