I am studying Statistics for data science since few months..
1)I am learning that, when we have to compare multiple samples (>2) then a T test would be tedious and instead we go for ANOVA and conduct a 'F Test'.
2)Above understanding kind of creates a 'mutually exclusive requirement between F test and T test."
3)I have also learned that, the T test ( be it : 1 sample/paired/2 sample ) basically tests for differences in means whereas 'F test' tests for differences in Variances.
4)Now, suppose two samples groups are having nearly equal means but highly different variances, then, both tests would give different answers right?
T test would say 'they are not different'. But 'F test' would say 'they are different'.
Or even for a reverse case. (hugely different means, but nearly same variances)..
5)So based on what, (the mean? or the variance?) we are finally going to decide their true difference?
6)So Question is: How are they related? If original objective was to find out two or more samples are different or no, then how 'looking for means'(i.e. choosing the T test) for smaller no of sample groups, gets changed to 'looking for variances' when no of sample groups are >2? (When the fact is :the variance and mean are basically independent characteristics of a sample group)
7)Should Not these both metrics be checked for finding whether truly the two samples are different or Not ?
( I have mentioned serial numbers to points that I have stated. Kindly point out if any of them is a basically wrong understanding. Would appreciate if answers are given for each point)