How to test a treatment had a positive effect on the experimental group? For the experiment, 12 people were invited. All participants were pre-tested (10 tasks with 0/1 marks). Then we randomly divided all the participants into two equal groups of 6 people: experimental and control.
A treatment was conducted and all participants were post-tested (again 10 task with 0/1 marks).
Results of tests:
data = {'e0':  [6, 5, 5, 5, 6, 0, 0, 0, 1, 0], # experimental group, before a treatment 
        'c0':  [6, 5, 5, 5, 2, 2, 1, 0, 2, 0], # control group, before a treatment 
        'e1':  [6, 6, 6, 6, 6, 5, 2, 6, 2, 2], # experimental group, after a treatment 
        'c1':  [6, 6, 0, 6, 0, 2, 3, 3, 4, 2]} # control group, after a treatment 

It is required to test a treatment had a positive effect on the experimental group.
Question. What is a statistical test suitable here?
My attempt is:
$H_0: \mu_0 = \mu_1$ vs.
$H_a: \mu_0 < \mu_1,$ where the $\mu_i$ are the respective means for pre- and post-tests in the experimental group.
But in hypothesis statement I don't use the control group results.

Because it's a small sample, this question and answer(s) are relevant. You're correct that you don't want to conduct a paired samples t-test on only the experiment group, because a comparison to a control group is important to your inference.
You could conduct an independent samples t-test on the post-task scores which would mean that the $H_0$ is the control group and $H_1$ is the experiment group. However, that does not leverage the baseline scores so if the groups randomly differed at baseline it could lead to faulty conclusions.
One option would be to calculate a difference score for each participant and conduct an independent samples t-test on the difference scores between the two groups.
There are a variety of issues to consider, in the linked post as well as other similar resources that address the assumptions of the t-test so I will not elaborate here. But there are certain conditions that you would want to check in order to be confident in the results of the t-test.
Regardless, this is a very small sample for a between-groups experimental design so it would be good to interpret the results cautiously.
