110

J. M. Gavilan, F. J. Ortega

degrees of freedom). Therefore, the SCI for

µ

at confidence level

1

−

α

is given

by the ETCI

!

¯

X

−

t

1

−

α/

2

(

n

−

1)

S

√

n

,

¯

X

+

t

1

−

α/

2

(

n

−

1)

S

√

n

"

,

where

t

1

−

α/

2

(

n

−

1)

is the quantile of order

1

−

α/

2

of the distribution

t

(

n

−

1)

.

For the particular case of the sample of size

n

= 10

in Table 1, which has been

generated from a normal distribution with

µ

= 0

and

σ

= 5

using, in software R,

the function rnorm() with seed equal to 100, the SCI and ETCI at 95% confidence

obtained is (-2.0968, 1.9172) whose length is 4.0140. Figure 1 illustrates that,

indeed, this CI is the SCI for

µ

based on the considered pivot.

Table 1: Random sample of size

n

= 10 generated from a centred normal distri-

bution with

σ

= 5

,

using in software R, the function rnorm() with seed equal to

100.

-2.5109618 0.6576558 -0.3945854 4.4339240 0.5848564

1.5931504 -2.9089534 3.5726636 -4.1262971 -1.7993107

Figure 1: Lengths of the CIs at 95% confidence for

µ

from the sample in Table

1 as a function of

α

1

, where

α

1

∈

[0

,

0

.

05] and

q

1

is the

α

1

-quantile of the

distribution

t

(

n

−

1) of the pivotal quantity

Q

, see (1.2)-(1.3).

0.00 0.01 0.02 0.03 0.04 0.05

4.0 4.5 5.0 5.5 6.0 6.5 7.0

With regard to the CIs involving scale parameters in normal populations,

the situation is different. These are based on pivotal quantities

Q

that follow

chi-squared distributions (for one population) and

F

distributions (for two pop-

ulations). These distributions are unimodal but not symmetrical. Therefore the