When conducting a test of hypothesis, we observe a sample and we make a statement about a restriction on the probability distribution that generated the sample. The hypothesis that the restriction is true is called null hypothesis.

The null hypothesis is usually denoted by .

In a test of the hypothesis that the mean of the distribution generating the sample is equal to zero, the null hypothesis is

The test can lead to either reject or not reject the null hypothesis. Rejection is decided upon based on the value of a test statistic that is constructed with the sample data at hand (see, e.g., the lecture entitled Hypothesis tests about the mean).

Note that failure to reject a null hypothesis does not constitute, per se, strong evidence that the null hypothesis is indeed true. For example, failure to reject may be a consequence of scarce availability of data. More generally, failure to reject could be due to the lack of statistical power of the test employed to test the null hypothesis. In other words, it could be due to the fact that the test of hypothesis had, ex ante, a low probability of rejecting a wrong hypothesis.

The lecture entitled Hypothesis testing provides a more detailed explanation of the term.

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