Classical statistics estimation of the Normal distribution mean when the standard deviation is known

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This derivation uses an important technique in classical statistics known as the pivotal method, which is composed of two stages:

1. Find a pivotal quantity, i.e. a random variable whose distribution does not depend on the parameter being estimated. The choice of the random variable is often based on finding a sufficient statisticA statistic Tn=tn({xi}) is called sufficient for estimating a parameter q if the conditional distribution of the data {xi} given Tn = some constant c does not depend on q for all values of c, i.e. once the value of Tn is known, {xi} contain no extra information about q.;

2. Invert (pivot) the inequality bounding the sample statistic into an inequality bounding the population parameter

The sample mean x of a set of n values {xi} is given by the formula:

For random samples from a Normal distribution, we have from Central Limit Theorem:

We have managed to create a relationship between x and m where they are independent of the probability distribution: the first stage of the pivotal method. The second stage is to rearrange the equation in terms of the parameter being estimated:

which is equivalent to the z-test, e.g. replace N(0,1) with z-scores for the 5% and 95% in the above formula and you have the 90% CI for m.

Alternatively, we could write:

which can be directly used in an ModelRisk model to simulate uncertainty about m.