Apart from what Matori just said, F test and Chi test can not be used in place of T or Z as T and Z are test of mean.
z test: normal distribution, population variance known, population variance unknown if sample size>30, test of mean such as avg=3, avg!=3, avg>3 or avg<3
t test: normal distribution, population variance unknown, sample size large or small, test of mean such as avg=3, avg!=3, avg>3 or avg<3
t test: comparison of means avg1=avg2
if population means are assumed to be equal and independent, t test with sp variance is used wherein sp is calculated using both s1 and s2
if population means are not assumed to be equal and independent, t test with s1 and s2 formula is used.
Paired comparison test: test for avg1=avg2 when means are dependent such as stocks from the same industry.
Chi test: var=3 var!=3, var=0 etc etc Remember this is a variance test, not mean.
F test: var1=var2
All of the above tests assume that you are testing for normal distribution. If you have non normal distribution, distribution with large sample size can be considered normal. Therefore, all tests for normal distribution can be applied to non normal distribution as long as sample size is large(30) enough. If sample size is not large(<30), non parametric test need be applied. I hope it helps!