Conclusions of Hypothesis Testing

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A general hypothesis is defined as following (eg a hypothesis on the population mean):

H0: Mu = Mu0
H1: Mu !=  Mu0

OK, apart from we have a two or one sided hypothesis, after performing the checking and statistical tests: our conclusion should be one of the following:
  • Rejecting the null hypothesis (H0).
  • Failing to reject the null hypothesis (H0).
The following statements for conclusions are not accurate:
  • Accepting the null hypothesis (H0).
  • Accepting the alternative hypothesis (H1).



But why?

When we fail to reject H0, it does not mean we accept H0 as a fact because we still could not prove it as a fact. But what happened is that we failed to prove it to be false. This goes like following: we have suspected new factors may affected the population mean, then we have taken all possible evidences and checking, but all checking failed to prove our suspects.

As well, rejecting H0 does not mean accepting H1 as a fact. What happens in this case is we prove, statistically, that H0 is false but not necessary H1 is true fact. Simply: our evidences and checks for the mean proved that it has changed, but we still have no guarantee that it changed into H1 region or this was due to different reasons/factors.
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