12.4 Research Hypothesis as Null Hypothesis

As noted before (Section ??), the research hypothesis usually is the alternative hypothesis. We expect something to change, to be(come) different rather than be or stay the same. We expect an association to be present rather than absent.

In this situation, rejection of the null hypothesis, which is the nil, supports our alternative hypothesis, hence our research hypothesis, so we are glad if we reject the null hypothesis. Of course, we know that we can be wrong. Our null hypothesis may still be true even if the probability of drawing a sample like the one we have drawn is so small that we have to reject the null hypothesis. This is a Type I error.

Fortunately, we know the probability of making this error because it is the significance level that we have chosen, five per cent usually. We can live with this probability of making an error if we reject the null hypothesis. So we are doubly glad: We found support for our research hypothesis and we know how confident we are about this support.

What if our research hypothesis is our null hypothesis? For example, we have a specific idea of average candy weight in the population from previous research or from specifications by the candy factory. Let us say that average candy weight is 2.8 grams according to the factory.

If we want to test whether the candies have the specified average weight, our research hypothesis would specify this average weight: Do candies weigh on average 2.8 grams in the population? Specifying a particular value, the research hypothesis must be the null hypothesis (Section ??). In this example, H₀: Average candy weight is 2.8 grams in the population.

If the research hypothesis is the null hypothesis because it contains a single (two-sided) or boundary (one-sided) value for the population parameter, we find support for our research hypothesis if we do not reject the null hypothesis. We can be wrong in not rejecting the null hypothesis. If we do not reject a null hypothesis that is actually false, we make a Type II error.

The significance level is irrelevant now because the significance level is the probability of making a Type I error. We do not reject the null hypothesis, so we can never reject a true null hypothesis (Type I error). Instead, the probability of making a Type II error is important, or rather, the probability of not making this error. This is the power of the test.

So if our research hypothesis represents the null hypothesis and our research hypothesis is supported (not rejected), we need test power to know how confident we can be about the support that we have found. Here, test power is relevant, not statistical significance.