Assume my hypothesis is: "Light supports the performance of students." The experimental setting consists of two room conditions that students are randomized to. The "light" group takes a test in a well lit room and the "dark" group has to take the same test in almost complete darkness (This sounds like a great idea for an experiment!). For each group, the test consists of 5 components: listening, speaking, presenting, translating and writing.
Surprisingly, the outcome is that the light experimental group outperformed the dark group in the total score. But it turns out that the results of the single test components differ to some extent; i.e. there is almost no difference as to listening, speaking, and presenting, but with regard to translating and writing there is some clear divergence. Below you find some numbers to help you imagine the situation.
Assuming that the general probability value is p < .05 for this experiment, the p-values of the single components cannot be interpreted without an alpha correction due to multiple testing, as far as I know. Applying a Bonferroni correction creates a new p < .01, but then there is insufficient statistical evidence to reject the null hypothesis for the tasks of writing and translating.
So here is my idea: I would like to combine the two tasks to a cluster with the explanation that for both of them you need to be able to see what you are writing. In other words, I want to create a kind of a posteriori hypothesis: "Light supports the performance of students in written tasks."
In order to get statistical significance I would like to adjust the alpha value with the Bonferroni correction like this: alpha = .05/5*2
Does this makes sense and is it a sound procedure?
Many thanks for any advice in advance