Super Impressive Bread Maker (Bonferroni Correction)

Suppose that I have a bakery store. One day, a guy came over to promote his bread-making machine. He emphasized that his machine can produce softer breads and crisper crusts. He allowed me to test it.

So I need to establish two hypothesis testing procedures:

1. The first test is to know whether his machine gives a softer inside part of the bread.
2. The second test is to see whether his machine gives bread with crisper crusts.

You know, buying a machine costs me money. So the machine must be very impressive in order for me to purchase the product. So for sure, I will use a significance level . However, that is not enough. I need to use the Bonferroni correction because I’m testing two hypotheses at the same time.

So what’s Bonferroni correction? It is a technique to counteract the problem of multiple comparisons. Suppose that we have two hypotheses to test, then the likelihood of observing a rare event increases. Therefore, the possibility of incorrectly rejecting a null hypothesis, i.e., the likelihood of committing a type I error, increases. This is because if we have two events and , then

Here the family-wise error rate is the probability of rejecting at least one true hypothesis. So the Bonferroni correction counters this issue by using a significance level of for each of the tests. Here we use a significance level for the whole two tests. So if Bonferroni correction is used, we will use a significance level for each individual test.

In a similar manner, if I test three hypotheses at the same time, then using a Bonferroni correction, we will use a significance level for each individual test.