The null hypothesis, you familiarly call it H_{0}, has a misleading name. Despite what one might think, that improper name doesn’t prevent it to be the core of all hypothesis testing.

And, what is hypothesis testing?. Let us see an example.

Let us suppose we want to know if residents (as they believe) are smarter than attending physicians. We pick out a random sample composed by 30 assistants and 30 residents from our hospital and we measure their IQ. We come up with an average value of 110 for assistants and 98 for residents (sorry, I’m an assistant and, as it happens, I’m writing this example). In view of these results we ask ourselves: what is the probability that the group of assistants selected are smarter than the residents of our example?. The answer is simple: 100% (of course, provided that everyone have passed an intelligence test and not a satisfaction survey). But the problem is that we are interested in knowing if assistant physicians (in overall) are smarter than residents (in overall). We have only measured the IQ of 60 people and, of course, we want to know what happens in the general population.

At this point we consider two hypotheses:

1. The two groups are equally intelligent (this example is pure fiction) and the differences that we have found are due to chance (random). This, ladies and gentlemen, is the null hypothesis or H_{0}. We state it in this way:

H_{0}: CI_{A} = CI_{R}

2. Actually, the two groups are not equally intelligent. This will be the alternative hypothesis:

H_{1}: CI_{A} ≠ CI_{R}

We could have stated this hypothesis in a different way, considering that IQ from one people being greater o smaller than other people’s, but let’s leave it this way for now.

At first, we always assume that H_{0} is true (and they call it null), so when we run our statistical software and compare the two means we come up with a statistical parameter (which one depend on the test we use) with the probability that differences observed are due to chance (the famous p). If we get a p lower than 0.05 (this is the value usually chosen by convention) we can say that the probability that H_{0} is true is lower than 5%, so we reject the null hypothesis. Let’s suppose that we do the test and come up with a p = 0.02. We’ll draw the conclusion that it is not true that both groups are equally clever and that the observed difference is not due to chance (in this case the result was evident from the beginning, but in other scenarios it wouldn’t be so clear).

And what happens if p is greater than 0.05?. Does it mean that the null hypothesis is true?. Well, maybe yes, maybe no. All that we can say is that the study is no powerful enough to reject the null hypothesis. But if we accept it as true without further considerations we will run the risk of blunder committing a type II error. But that’s another story…