Can a hypothesis test be performed if I have a non-normal population, small sample size, but population standard deviation is known? We are testing if the mean differs from the given mean.
The t-test is quite robust to departures from the assumption of normality. Intuitively, the reason for this is that the T statistic is based on averages, which are asymptotically normal under mild conditions (Central limit theorem), and typically they converge fast to normality.
Alternative nonparametric tests (for equality of distributions or means) include the Kolmogorov-Smirnov test, permutation tests, and Wilcoxon signed-rank test.
I think when the assumption of normality is not met, you should do non-parametric tests of significance, if you had variables like rankings or categories. You could still continue with t-tests for the mean if you have a sample size above 30.