In a test of hypothesis, a sample is observed and a statement is made about a restriction on the probability distribution that generated the observed sample.

The hypothesis that the restriction is indeed true is called null hypothesis, while the hypothesis that the restriction is not true (because another restriction incompatible with it is true) is called alternative hypothesis.

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Usually, the null hypothesis is denoted by , while the alternative hypothesis is denoted by .

Suppose the observed sample is made of draws from a probability distribution having mean . Suppose we want to test the hypothesis that the mean is equal to zero. Then, the null hypothesis can be written as An alternative hypothesis is another hypothesis incompatible with . For example, if any other value of the mean is possible, then the alternative hypothesis isInstead, if the mean can take only non-negative values, we have

The formulation and use of alternative hypotheses in statistical hypothesis testing is controversial and there seems to be no unanimous agreement on the exact meaning of the term alternative hypothesis.

A statistical test results in two possible outcomes: either the null hypothesis is rejected or it is not rejected. According to some statisticians, rejecting the null is equivalent to accepting the alternative. However, others deem that rejecting the null does not necessarily imply accepting the alternative, because it is possible to think of situations in which both hypotheses can be rejected (for example, when a mis-specified statistical model is being used to conduct the test).

You can find a longer discussion of the controversy surrounding alternative hypotheses, as well as a more detailed explanation of the concept, in the lecture entitled Hypothesis testing.

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