I am working on a project about a diagnosis system where I have two scenarios.

In the first I have binary data like:

highTemp

0

1

etc.

where zero indicate normal temperature and 1 indicate high temperature. I want to correlate this with a class whose values can be one of three mechanic systems (dependent variable) that I encoded 1, 2, and 3. Which test should I use in this case? I thought about chi square but my data (highTemp) is not normally distributed.

In the second problem, I just wanted to know how likely is to see an independent binary variable like highTemp occur when we have another independent binary variable like engine speed (encoded as 0 for normal speed and 1 for too fast). Here as well, my both binary variables are not normally distributed according to Shapiro-Wilk test. Would Chi-square work here? Chi-square is the only test I could see that does not require a dependent and independent variables, but it does require normal distribution though.

Your help is highly appreciated