Which sampling distribution is most appropriate for estimating a mean when the standard error of the mean is NOT known

Answer 1

It's the t-distribution. That's it. t.

OK, more detail. The t-distribution, also known as Student's-t distribution has many of the desirable properties of the normal distribution: it's symmetrical, roughly bell-shaped, and is applicable to many of the same problems. It has one big advantage: you don't have to pretend you know the true standard error of the mean. Just like with the normal distribution, you calculate the standard error from the data, but, unlike the normal distribution, you don't pretend you already knew it. For small samples, a t-distribution and a normal distribution look similar, like siblings. However, as the sample gets bigger and bigger, the t-distribution and the normal distribution look more like twins.

Some added advantages, whose detail is beyond the scope of this answer: the t-distribution is extremely useful in many simple or challenging hypothesis tests. Also, the t-distribution is related to both the F-distribution and the chi-square distribution which, with the t and normal distributions make up the most commonly used distributions.