Symmetric hypotheses
In some situations there is a kind of symmetry between two competing hypotheses. For example, if two candidates, A and B, stand in an election and π is the population proportion who will vote for A, we are interested in which candidate will win:
H1 : π > 0.5
H2 : π < 0.5
Null and alternative hypotheses
In statistical hypothesis testing, the two hypotheses are not treated symmetrically in this way. Instead, we ask whether the sample data are consistent with one particular hypothesis (the null hypothesis, denoted by H0). If the data are not consistent with H0, then we can conclude that the competing hypothesis (the alternative hypothesis, denoted by HA) must be true.
The two possibilities are:
We should never conclude that H0 is likely to be true.
Example
Consider a test for whether a population mean is zero:
H0 : µ = 0.0
HA : µ ≠ 0.0
Based on a random sample, we might conclude: