Classical statistics estimation of the Normal distribution mean when the standard deviation is not known

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From the derivation of the mean estimate for a Normal distribution when s is known we had:

                            (1)

However, we don't know s. From the derivation of the standard deviation when the mean is unknown, we had:

                                 (2)

Since the uncertainty distributions for m and s are independent we can combine Equations (1) and (2):

          (3)

where

is the unbiased estimate of the standard deviation given by Excel's STDEV function. The component in the square brackets of Equation (3) is actually a Student-t distribution:

so the Equation (3) for m reduces to:

where t(n-1) is a Student-t distribution with (n-1) degrees of freedom. This last equation is directly used in this ModelRisk example model to simulate uncertainty about m.