Can a t-test/ANOVA approach to hypothesis testing be carried out when data has well-defined upper and lower limits and is not continuous?
Let's assume a situation where data are generated by some phenomenon resembling some Gaussian/normal distribution (i.e. $ X \sim N(\mu,\sigma) $), but such that the data is integer-valued (i.e. $ X \in Z $ ) and that the data lies within a finite range (i.e. $ [a, b], \infty < a < b < \infty $).
Obviously, there are a few differences between the distribution of the values generated by the phenomenon and a Gaussian/normal distribution:
- $ X $ is not continuous.
- $ X $ does not have a support that extends into infinity in either direction.
However, the data generated by this distribution will still look kind of Gaussian/normal in many ways.
(a) Now, assume we have a similar phenomenon in which data are generated in a seemingly similar but unknown process.
We want to compare the distributions of both phenomena to see if there is a statistically significant difference between the phenomena distributions.
Can we use a t-test to perform this analysis?
(b) Now, assume we have several similar phenomena which generate data in a seemingly similar but unknown process.
We want to compare the distributions of all the phenomena to see if there is a statistically significant difference between at least one of the phenomena distributions and the other distributions.
Can we use ANOVA to perform this analysis?
(EDIT: I have changed the framing of this question in order to better capture the issue)