How to test significance in questionnaire answers from 0 to 10

Question

I am currently generating a panel dataset with random numbers. I simulate a questionnaire where "people" get to answer on a scale of 0 to 10 with 10 being the best. Creating the data was neasy. I used this to get answers where I assume that the average answer would be around 7.

variable <- rbinom(n, 10, 0.70)

Now in the panel data I want to create fluctuations in later periods, which is also not too difficult jut changing the probability parameter and then compare them to a historical mean to infer whether the change in wellbeing is statistically different.

So my question is: What test do I conduct hypothetically in a real scenario? Since people can only answer from 0 to 10 it is not really normally distributed, however I can choose n to be very high. T-test? And what about confidence intervals?

I know that the data is not continuous and that a mean of 6.942 doesn't exist because it is not defined. I neglect this basing my decision on a paper I read. There they choose to treat the scores as contiuous variables for simplification purposes. I have read in other threads that there are super sophisticated approaches for evaluating this kind of stuff. The paper then shows graphs where certain events triggered a decline in wellbeing that was statistically significant but the version I have does not go into detail on how they conducted the significance test.

Any help is massively appreciated!

Answer 1

What do you want to know from the test, eg. what is the specific question you are asking? If the mean is different ?

Perhaps the Mann-Whitney U test would suffice if you feel mean's aren't really of value to you? It answers a slightly different question: "Would you expect one population to have higher values than another", which still sounds applicable to your case.

set.seed( 100 )
n <- 50
ref.data <- rbinom(n, 10, 0.70)
new.data <- rbinom(n, 10, 0.60)
wilcox.test( ref.data, new.data, paired=FALSE )