For example, in order to decide whether a dice is fair, that is, unbiased, a hypothesis can be made that a particular number, say 5, should occur with a probability of one in six, since there are six numbers on a dice. Such a hypothesis is called a null hypothesis and is an initial statement. The symbol H0 is used to indicate a null hypothesis. Thus, if p is the probability of throwing a 5, then H0 : p = 16 means, ‘the null hypothesis that the probability of throwing a 5 is 16 ’. Any hypothesis which differs from a given hypothesis is called an alternative hypothesis, and is indicated by the symbol H1. Thus, if after many trials, it is found that the dice is biased and that a 5 only occurs, on average, one in every seven throws, then several alternative hypotheses may be formulated. For example: H1: p = 17 or H1: p < 16 or H1: p > 18 or H1: p = 16 are all possible alternative hypotheses to the null hypothesis that p = 16
Hypotheses may also be used when comparisons are being made. If we wish to compare, say, the strength of two metals, a null hypothesis may be formulated that there is no difference between the strengths of the two metals. If the forces that the two metals can withstand are F1 and F2, then the null hypothesis is H0 : F1 = F2. If it is found that the null hypothesis has to be rejected, that is, that the strengths of the two metals are not the same, then the alternative hypotheses could be of several forms. For example, H1 : F1 > F2 or H1 : F2 > F1 or H1 : F1 = F2. These are all alternative hypotheses to the original null hypothesis.