Editing Null hypothesis (section)

====Tailedness of the null-hypothesis test====
Consider the question of whether a tossed coin is fair (i.e. that on average it lands heads up 50% of the time) and an experiment where you toss the coin 5 times.
A possible result of the experiment that we consider here is 5 heads. Let outcomes be considered unlikely with respect to an assumed distribution if their probability is lower than a significance threshold of 0.05.

A potential null hypothesis implying a one-tailed test is "this coin is not biased toward heads". Beware that, in this context, the term "one-tailed" does ''not'' refer to the outcome of a single coin toss (i.e., whether or not the coin comes up "tails" instead of "heads"); the term "[[One- and two-tailed tests|one-tailed]]" refers to a specific way of testing the null hypothesis in which the critical region (also known as "[[Statistical hypothesis testing#Definition of terms|region of rejection]]") ends up in on only one side of the probability distribution.

Indeed, with a fair coin the probability of this experiment outcome is 1/2<sup>5</sup> = 0.031, which would be even lower if the coin were biased in favour of tails. 
Therefore, the observations are not likely enough for the null hypothesis to hold, and the test refutes it. 
Since the coin is ostensibly neither fair nor biased toward tails, the conclusion of the experiment is that the coin is biased towards heads.

Alternatively, a null hypothesis implying a two-tailed test is "this coin is fair". 
This one null hypothesis could be examined by looking out for either too many tails or too many heads in the experiments.
The outcomes that would tend to refute this null hypothesis are those with a large number of heads or a large number of tails, and our experiment with 5 heads would seem to belong to this class.

However, the probability of 5 tosses of the same kind, irrespective of whether these are head or tails, is twice as much as that of the 5-head occurrence singly considered.
Hence, under this two-tailed null hypothesis, the observation receives a [[P-value|probability value]] of 0.063.
Hence again, with the same significance threshold used for the one-tailed test (0.05), the same outcome is not statistically significant.
Therefore, the two-tailed null hypothesis will be preserved in this case, not supporting the conclusion reached with the single-tailed null hypothesis, that the coin is biased towards heads.

This example illustrates that the conclusion reached from a statistical test may depend on the precise formulation of the null and alternative hypotheses.