I have got 2 large samples and want to compare the difference. There are 2 methods:
- Assume the population variances are unknown and unequal, and use the Welch $t$ test.
- Assume the population variances are known and unequal (estimated as sample variances, which is ok since my sample size is million level and the sample mean can be considered as normally distributed), and use the $Z$ test.
I found these 2 tests are pretty much the same formula and same result. So are these 2 tests the same when applied to the real world situation? And if not, what is the difference and when should I use the $Z$ test/welch $t$ test?