I want to conduct an experiment.
I am checking a vaccine, i want to see if the side effects to the vaccine can predict the immunogenicity of the vaccine.
I take 2 groups, 1 that after the vaccination had mild side effects, the other had serious side effects.
I will estimate the immunogenicity of the vaccine by the number of antibodies in the blood of the participants after the vaccination.
I want help in thinking which test i should use.
$H_0$: There is no difference in the immunogenicity (using mean number of antibodies in each group - $\mu$)-> $\mu_0 = \mu_1$
$H_1:$ There is a difference, and the immunogenicity in the group that has worse side effects is bigger (using mean number of antibodies - the mean number of antibodies in the group that had worse side effects is bigger)-> $\mu_0 < \mu_1$
The numbers of participants in each group will be greater than 30: $n > 30$
Therefore, i though to use $Z$ (Normal distribution) Test.
The problem: I dont know the mean In the population, i must estimate that from the group.
So, can i use $Z$ Test when i dont know the mean in the population?
Otherewise -> what should i use? Or how to estimate the mean?
Thanks.