t-tests and p-values

We have discussed how to handle this test when σ is known. But how should we proceed if it is unknown? One option is the GLRT, discussed above. However, (a) we must estimate μ and (b) Wilk’s theorem only tells us that the test statistic corresponding to maximum likelihood estimates of σ and μ is asymptotically chi-squared. For small n, then, it can be difficult to set a threshold to achieve a desired probability of false positives or type I error. As alternative is the celebrated t-test. Specifically, let