Fundamentals of Probability and Statistics for Engineers

.R esult i: let^2 be the unbiased estimator for^2 as defined by Equation

(11.33), and let

It follows from the results given in Section 9.3.2.3 that D is a^2 -distributed

random variable with (n 2) degrees of fr eedom.

.R esult ii: consider random variables

^

^

y = ^+^x

10 20

0

20

40

60

80

100

120

140

Young's modulus,

y

(g/cm

2 )

Stress,x (g/cm^2 )

Figure 11.3 Estimated regression line and observed data, for Example 11.3

348 Fundamentals of Probability and Statistics for Engineers

c 

Dˆ

…n 2 †c^2

^2

: … 11 : 34 †



…A^ † c^2

Xn

iˆ 1

x^2 i

"

n

Xn

iˆ 1

…xix†^2

(^8) "# 1
<
:

9

=

;

 1 = 2

; … 11 : 35 †

σ

^

α β

σ