I am working with a dataset with only 8 subjects. I am working with a rare event so increasing the sample size is not a possibility. I was wondering if I can do a one sample T-test with this data if the normality assumption is satisfied? Or a Wilcoxon one sample signed rank test, if not? In addition, does it make sense to do a test (parametric or nonparametric) with only 8 subjects? Thank you in advance for your response!
Of course you can do a test with 8 observations.
The reason is that tests of significance were designed and developed for small samples. Before that, scientists had to collect very large samples so they could look at the descriptive statistics and histograms and say things like "Hmm, looks pretty different to me." Then their friend(?) down the hall would say something like "Hmmm, I'm not sure. Looks like some funny outliers there."
Tests of significance permitted scientists to come to conclusions with small samples.
This was Student's (true name, WS Gossett) problem as a chemist for Guinness Breweries. He took samples of stuff, tested it in the lab, and then did statistical comparisons. Doing lab work used up his time and he could often only get small samples.
Some samples are too small, of course. I think five is about as low as you can go because (1/2)^5 is 1/32 or just smaller than .05 but (1/2)^4 is 1/16 or just larger than .05. But then .05 isn't so important anyway so maybe n = 4 is about as low as you can go (since (1/2)^3 = 1/8) and still use statistics.
Likert scale data is problematic for a lot of standard statistical techniques, but in your case the sample size is probably too small to resolve the discreteness of your scale. It's probably okay to do a t-test, but I would probably choose the more non-parametric sign test.