I know this is a very debated topic, even on this site, but I still couldn't find an answer to my problem.
Recently I am working with large samples (300, 400 and more). For now, I am trying to use simple techniques, such as correlation, T-tests, and ANOVA, all of which require the normality assumption (from what I have read so far in the textbooks, online etc.) I also read that, if the sample size is large enough, the normality assumption is not so much of a problem and these techniques are robust to the violation of normality. Is normality a problem given my sample size? Should the data be at least bell-shaped, even if the tests fail to accept normality? Or could I get away even with extremely skewed or lumped data when using these techniques?
Should I apply parametric techniques or should I just stick to the non-parametric ones, which, from what I know, have lower power?
LATER EDIT for those who want to find more about my data:
I have a variable which represents the number of days a user has been employed in the program (mean=176, median=167, stdev=87, IQR=113, Skewness=0.61, kurtosis=-1.64) sample=340users. The histogram does show that maybe the variable has the potential to split into 'fairly normal' groups (however I have not found this factor).
This is the variable that for now i am trying to explain, in terms of 'The Number of Weeks The User has worked in the first 8 weeks', which takes values from 1 to 8, so I assume it is ordinal, and has a negative skew(most of them have worked 8 weeks out of 8). So, the main question would be is there a relationship between how much a user will stay employed and the amount of work he does in the first 8 weeks?
Also, later I would compare the length of employment with other possible factors, for which currently I don't have the data (education, gender, age, ..)and try to build a more 'elaborate' model, but right now I try to analyse what I have with 'simple' statistics.
For now, I have done a Spearman correlation between length of employment and weeks worked out of first 8, had a coefficient of 0.450 (which means low correlation), so I am trying to see if those who have worked an amount of no of weeks in the first 8 differentiate themselves in terms of length of employment from those who have worked fewer/more weeks. So I studied the distributions of length of employment for each of the groups (a group means a certain no of weeks they have worked in the first 8). In each group I have around 25-40 cases (except weeks worked=0 where I have 9cases and weeks worked=9 where there are 156). The normality test (Shapiro-Wilk, which I knew were suitable with such sample size) showed that only 2 groups out of 9 are normally distributed. So my thought was to drop ANOVA and t-tests and head for Mann-Whitney U Test and Kruskal-Wallis Test. However now I am re-considering maybe looking at ANOVA and t-tests because the groups do not look that non-normal. Thanks.