The Shortest Confidence Interval for the Mean of a Normal Distribution

An interesting topic in mathematical statistics is that of the construction of the confidence intervals. Two kinds of intervals which are both based on the method of pivotal quantity are the shortest confidence interval and the equal tail confidence intervals. The aim of this paper is to clarify and comment on the finding of such intervals and to investigation the relation between the two kinds of intervals. In particular, we will give a construction technique of the shortest confidence intervals for the mean of the standard normal distribution. Examples illustrating the use of this technique are given.

Let ( 1 ,  2 , … ,   ; ) be a pivotal quantity where  1 ,  2 , … ,   is a random from the distribution of (; ).The probability statement is converted (when possible) to If constants  1 ,  2 in (1.1) can be found so that ( * 2 −  * 1 ) is minimum, then the interval [ * 1 ,  * 2 ] is said to be the shortest confidence interval based on .On the other hand if constants  1 ,  2 in (1.1) can be determined so that is said to be an equal tails confidence interval.The aim of this work is to clarify and comment on problems that emerge at the process of finding, to investigate the relation of equality of length of these.In particular, we will give a construction technique of the shortest confidence intervals for the mean of the standard normal distribution.It turns out that the symmetric solution  = − is optimal here.The symmetric solution is This result generalizes to any sampling distribution that is unimodal.
Theorem 1: Let  be a unimodal probability density function.If interval [; ] satisfies: is the shortest of all intervals that satisfy (i).
Let () be a positive statistic.Suppose that Proof: see references. ∎

Results
Suppose throughout this part: Let  be a random variable such that  ↝ ( = (),  2 ) (the normal distribution) and  1 ,  2 , … ,   a random  − sample of  with  2 known.We then know that a good point estimator of  is  ̅ .
Problem is: choose [; ] so that (, ) =  −  is the shortest length possible for a given confidence coefficient (1 − ).

Calculation of 𝑎 and 𝑏
The following result provides a general method of finding confidence intervals and covers most cases in practice.