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The alternative hypothesis (or maintained hypothesis or research hypothesis) and the null hypothesis are the two rival hypotheses whose likelihoods are compared by a statistical hypothesis test. Usually the alternative hypothesis is the possibility that an observed effect is genuine and the null hypothesis is the rival possibility that it has resulted from random chance.

The classical (or frequentist) approach is to calculate the probability that the observed effect (or one more extreme) will occur if the null hypothesis is true. If this value (sometimes called the "p-value") is small then the result is called statistically significant and the null hypothesis is rejected in favour of the alternative hypothesis. If not, then the null hypothesis is not rejected. Incorrectly rejecting the null hypothesis is a Type I error; incorrectly failing to reject it is a Type II error.

Bayesian statisticians would challenge this approach in that it takes no account of a priori beliefs in the two hypotheses or the different consequences of taking a wrong decision; there may be good reasons (extraneous to the statistical data) for believing the null hypothesis to be correct. This must be weighed against the damning evidence of a low p-value before the null hypothesis can be rejected.

An example: In the trial of Sally Clark, a solicitor accused of killing both her babies, pediatrician Sir Roy Meadow testified that the probability of two infants in the same family dying of natural causes was 1 in 73,000,000. If natural death is the null hypothesis and murder the alternative hypothesis, then the p-value is 1/73,000,000. The smallness of this value means that the null hypothesis that the deaths had had natural causes should be rejected and therefore murder concluded.

The problem was that even if the 73,000,000 figure were correct (this calculation was itself challenged as being flawed by the ecological fallacy), double murder is a rare event and there is therefore a good a priori reason for believing the null hypothesis. The standard hypothesis test was therefore not a good indicator of Clark's guilt.

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