Fundamentals of Probability and Statistics for Engineers

H ence,^2 is unbiased with k 1/(n 2), giving

or, in view of Equation (11.30),

Example 11.2.Problem: use the results given in Example 11.1 and determine
an unbiased estimate for^2.
Answer: we have found in Example 11.1 that

In addition, we easily obtain

Equation (11.33) thus gives

Example 11.3.Problem: an experiment on lung tissue elasticity as a function
of lung expansion properties is performed, and the measurements given in
Table 11.2 are those of the tissue’s Young’s modulus (Y), in gcm^2 ,atvarying
values of lung expansion in terms of stress (x), in gcm^2. Assuming that E Y
is linearly related to x and that^2 Y^2 (a constant), determine the least-square
estimates of the regression coefficients and an unbiased estimate of^2.

Table 11.2 Young’s modulus, y (g cm^2 ), with stress, x (g cm^2 ), for Example 11.3

x22.535791012151617181920
y 9.1 19.2 18.0 31.3 40.9 32.0 54.3 49.1 73.0 91.0 79.0 68.0 110.5 130.8

346 Fundamentals of Probability and Statistics for Engineers

c ˆ

c^2 ˆ^1

n 2

Xn

iˆ 1

‰Yi…A^‡Bx^i†Š^2 ; … 11 : 32 †

c (^2) ˆ^1
n 2
Xn
iˆ 1
…YiY†^2 B^^2
Xn
iˆ 1
…xix†^2

"

: … 11 : 33 †



Xn

iˆ 1

…xix†^2 ˆ 2062 : 5 ;

^ˆ 0 : 57 :

Xn

iˆ 1

…yiy†^2 ˆ 680 : 5 :

b^2 ˆ^1

8

‰ 680 : 5 … 0 : 57 †^2 … 2062 : 5 †Š

ˆ 1 : 30 :



 fg

 ˆ