A Note on Optimal Intervals in Normal Populations

111

ETCI does not provide the SCI. In order to obtain the SCI based on the pivot

Q

,

q

1

and

q

2

should be chosen to minimize the length

ℓ

=

q

∗

2

−

q

∗

1

subject to the

constraint of maintaining a probability of coverage 1

−

α

, as given in (1.2). In

these two cases, the length of the intervals built on such a pivot has the following

structure

ℓ

=

q

∗

2

−

q

∗

1

=

k

!

1

q

2

−

1

q

1

"

,

(2.1)

where

k

depends only on the random sample. Therefore, the constrained op-

timization problem is equivalent to solving the system of nonlinear equations

given by (1.2) and the equation

q

2

1

f

(

q

1

) =

q

2

2

f

(

q

2

) (Ferentinos and Karakostas,

2006).

Let us consider the case of one normal population

X

∼

N

(

µ, σ

). To obtain

CIs for the variance

σ

2

, the pivot

Q

=

(

n

−

1)

S

2

σ

2

is usually utilized, which follows

a distribution

χ

2

(

n

−

1) (chi-squared with

n

−

1 degrees of freedom). In this

case, the CIs at level 1

−

α

are given by

!

(

n

−

1)

S

2

χ

2

1

−

α

2

(

n

−

1)

,

(

n

−

1)

S

2

χ

2

α

1

(

n

−

1)

"

,

where

χ

2

α

1

(

n

−

1) is the

α

1

-quantile of the distribution

χ

2

(

n

−

1) and

α

1

∈

[0

, α

]

with

α

1

+

α

2

=

α

. As aforementioned, the length of the interval is as in (2.1):

specifically, it is given by

ℓ

= (

n

−

1)

S

2

!

1

χ

2

α

1

(

n

−

1)

−

1

χ

2

1

−

α

2

(

n

−

1)

"

,

which can be minimized and tabulated for each

n

and

α

. This table could be

used in a classroom to obtain SCIs for given samples and for its comparison with

the corresponding ETCIs. Table 2 shows the values of

α

1

, in up to four decimal

places, to be chosen in order to obtain the SCIs at 95% confidence for several

sample sizes.

Table 2: Optimal values of

α

1

to obtain the SCI at 95% confidence for the

variance of a normal population according to the sample size, and ratios of the

lengths of the corresponding ETCIs and SCIs.

n

3

4

5

6

7

8

9

10

11

12

α

1

.0499 .0499 .0497 .0494 .0491 .0488 .0484 .0480 .0476 .0473

ℓ

ETCI

/ℓ

SCI

2.0192 1.6175 1.4472 1.3525 1.2918 1.2493 1.2179 1.1936 1.1743 1.1585

n

13

14

15

16

17

18

19

20

21

22

α

1

.0469 .0465 .0462 .0458 .0455 .0452 .0449 .0446 .0443 .0441

ℓ

ETCI

/ℓ

SCI

1.1454 1.1343 1.1248 1.1165 1.1093 1.1030 1.0973 1.0922 1.0877 1.0835