# Determining probability from categorical variable data

Whist reviewing some test data I am looking for advice on identifying the correct statistical tool(s) to help me answer two questions. The data I am looking at is from a cytotoxicity test, whereby a single categorical variable result is returned for one test. A "0" denotes no cytotoxic response, whilst a "4" denotes the highest possible cytotoxic response. The acceptance criteria is that the cytotoxicity must be equal to or less than 2. I have access to Minitab software for analysis.

I have two sets of data: before and after an event which caused a shift in cytotoxicity results. I want to answer two questions:

1.) Are the proportions of each category significantly different between the two data sets?

2.) What is the probability of producing a score >2 for each sample data set at some point in the future, assuming no other shifts in the process?

I believe I can answer the first question using the Chi-Square Goodness-of-Fit test. However, I feel like I'm clutching at straws for the second question. I am thinking about the binomial probability of success but this only works with attribute data (pass/fail) and not categorical.

A basic bar chart has been included below using some example data to illustrate my point. As you can see the population on the right has a "wider" distribution compared with the left. Assuming I had more data, can I produce a probability that I will receive a score of 3 or 4 at some point in the future?

OK, I think you only need a simple test of proportions where you are testing that $$H_0: p_0 = p_1$$ where $$p_i$$ denotes the population proportion for cytotoxic. In Minitab I believe this is called a two-sample proportion test.