Fundamentals of Probability and Statistics for Engineers

and

where, as seen from Equations (11.20), (11.22), and (11.23), and are,

(^2) estimated by (^2). The derivation
given in Section 9.3.2.2 shows that each of these random variables has a
t-distribution with (n 2) degrees of fr eedom.
. (^) Result iii: estimator for the mean of Y is normally distributed with
mean x and variance
H ence, again following the derivation given in Section 9.3.2.2, random variable
is also t-distributed with (n 2) degrees of freedom.
Based on the results presented above, we can now easily establish confidence
limits for all the parameters of interest. The results given below are a direct
consequence of the development in Section 9.3.2.
.R esult 1: a [100(1 )]% confidence interval for is determined by [see
Equation (9.141)]
Linear Models and Linear Regression 349
respectively, the means of and andthedenominatorsare, respectively,
thestandard deviationsof and with
…B^ † c^2
Xn
iˆ 1
…xix†^2
(^8) < "# 1

:

9

=

;

 1 = 2

… 11 : 36 †

A^ B^

A^ B^  c

EfYg

‡

varfEfdYggˆvarfA^‡Bx^g

ˆvarfA^g‡x^2 varfB^g‡ 2 xcovfA^;B^g

ˆ^2

Xn

iˆ 1

…xix†^2

"# 1

1

n

Xn

iˆ 1

x^2 i‡x^2  2 xx

!

ˆ^2

1

n

‡…xx†^2

Xn

iˆ 1

…xix†^2

(^8) "# 1
<
:

9

=

;

:

… 11 : 37 †

EdfYg… ‡ x†

hi

c^21

n

‡…xix†^2

Xn

iˆ 1

…xix†^2

(^8) "# 1
<
:

9

=

;

8

<

:

9

=

;

 1 = 2

… 11 : 38 †





L 1 ; 2 ˆA^tn 2 ;

= 2 c^2

Xn

iˆ 1

x^2 i

!

n

Xn

iˆ 1

…xix†^2

(^8) "# 1
<
:

9

=

;

1 = 2

: … 11 : 39 †

d