How do I chose the right distribution for this specific circumstance? Posted this on stack overflow and people said its probably more suited for this forum.
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 easy. I used this to get answers where I assume that the average answer would be around 7.
In R code
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?
So let's say, the baseline mean of average wellbeing is 7.0 (n=2000 individuals) Some periods in the future a natural catastrophe negatively influences wellbeing and the observations from this period show an average of 6.2. What test will I use to decide if this is significantly different from 7?
Any help is massively appreciated!
 A: I will simulate suitable data using R. The second sample tends to
have more high values.  (R converts the vector p
of proportions to probabilities that sum to $1.)$
set.seed(2021)
x1 = sample(0:10, 200, rep=T, p = c(1,1,1, 2,3,5, 5,4,3, 2,1))
x2 = sample(0:10, 200, rep=T, p = c(1,1,1, 2,6,7, 8,9,9, 9,9))

Boxplots give a graphical view of the two samples.
boxplot(x1, x2, horizontal=T, col="skyblue2")


Put the counts into a contingency table TAB.
t1 = tabulate(x1);  t1
[1] 16  6 12 25 34 37 33 21  8  4
t2 = tabulate(x2);  t2
[1]  2  3  6 17 17 34 32 24 29 31

TBL = rbind(t1,t2); TBL
    [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
t1   16    6   12   25   34   37   33   21    8     4
t2    2    3    6   17   17   34   32   24   29    31

A chi-squared test detects that the counts are not homogeneously
distributed in the two samples: P-value far below 5%. (Because some counts are low,
we invoke the feature of chisq.test in R, that simulates
a reasonably accurate P-value; an implementation of Fisher's
exact test to handle tables larger than $2\times 2$ would
be an alternate option--if your computer can handle it.)
chisq.test(TBL, sim=T)

        Pearson's Chi-squared test with simulated p-value 
        (based on 2000 replicates)

data:  TBL
X-squared = 54.167, df = NA, p-value = 0.0004998

Because the two distributions are roughly of the same shape,
a two-sample Wilcoxon rank sum test is another alternative.
Here sample sizes are large enough that the existence of many ties
does not trigger an error message in R's implementation of this test.
wilcox.test(x1, x2)

        Wilcoxon rank sum test 
      with continuity correction

data:  x1 and x2
W = 12778, p-value = 3.018e-10
alternative hypothesis: 
  true location shift is not equal to 0

Even if the two distributions were not nearly of the same shape,
the Wilcoxon test could be viewed as a test of stochastic domination.
The empirical CDF (ECDF) plots of the two samples show that the
second sample is plotted mainly to the right (hence below) the first, indicating its domination.

R code for ECDF plot:
hdr = "ECDF plots: Second Sample (solid blue) Dominates First"
plot(ecdf(x2), col="blue", lwd=2, main=hdr)
 lines(ecdf(x1), col="brown", lwd=2, lty="dotted")

