A Note on Optimal Intervals in Normal Populations

109

and then convert this relation into a relation as in (1.1) to obtain the CI for

θ

.

The length of the CI is given by

ℓ

=

q

∗

2

−

q

∗

1

. When, for a given

α

, it is not

possible to select another pair (

q

1

, q

2

) to reduce the length, then the CI is called

the shortest confidence interval (SCI) for

θ

based on

Q

and its length is denoted

by

ℓ

SCI

. On the other hand, if the quantities (

q

1

, q

2

) are chosen such that

P

(

Q < q

1

) =

α

1

and

P

(

Q > q

2

) =

α

2

,

(1.3)

where

α

1

∈

[0

, α

] and

α

1

+

α

2

=

α

, then many other CIs at level 1

−

α

are

obtained. In the particular case in which

α

1

=

α

2

=

α/

2

,

the CI is named the

equal-tails confidence interval (ETCI) for

θ

based on

Q

, and its length is denoted

by

ℓ

ETCI

. The ETCI and the SCI sometimes coincide. Specifically, this happens

provided that the distribution of the pivotal quantity is symmetric and concave

on the right-hand side of the point of symmetry (Kirmani, 1990); this also holds

true when the distribution of the pivot is symmetric and unimodal

1

(Casella and

Berger, 2002).

It is clear that, in general, the SCI is preferable to the ETCI since it is more

accurate, however only the ETCI is usually obtained in classrooms and it is the

only interval provided by the statistical software.

In this paper, the particular case of one and two normal populations is anal-

ysed, for which the pivots for the location and scale parameters and their corre-

sponding ETCIs are well-known. When the ETCI obtained is not the SCI, then

the difference in length between the two options is studied. The results can be

used in the classroom to obtain the ECTI in a straightforward way.

2. Confidence intervals in normal populations

In normal populations, the CIs involving location parameters are based

on pivotal quantities that follow a normal distribution when the scale parameters

are known, and on pivotal quantities that follow a

t

distribution when the scale

parameters are unknown. Therefore, taking into consideration that the normal

and the

t

distributions are symmetric and unimodal, the ETCIs are the SCIs.

Example 2.1. CI for the mean of one normal population.

Let us con-

sider a population

X

∼

N

(

µ, σ

)

(normal distribution of mean

µ

and standard

deviation

σ

). It is well known that the expression

t

=

¯

X

−

µ

S

√

n,

where

¯

X

is the sample mean and

S

the sample standard deviation, is a pivotal

quantity for

µ

which follows a distribution

t

(

n

−

1)

(distribution

t

with

n

−

1

1

A pdf

f

(

x

) is unimodal if there exists

x

∗

such that

f

(

x

) is non-decreasing for

x

≤

x

∗

and

f

(

x

) is non-increasing for

x

≥

x

∗

.