Page 363 - Fundamentals of Probability and Statistics for Engineers

346 Fundamentals of Probability and Statistics for Engineers

Hence, 2 is unbiased with k ˆ 1/(n 2), giving
c
n
1 X 2
^
^
c 2
ˆ ‰Y i A ‡ Bx i †Š ; 11:32†
n 2
iˆ1
or, in view of Equation (11.30),
" n n #
1 X 2 X 2
c 2
ˆ Y i Y† B ^ 2 x i x† : 11:33†
n 2
iˆ1 iˆ1
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

n
X 2
x i x† ˆ 2062:5;
iˆ1
^
ˆ 0:57:
In addition, we easily obtain

n
X 2
y i y† ˆ 680:5:
iˆ1
Equation (11.33) thus gives

1 2
ˆ ‰680:5 0:57† 2062:5†Š
b 2
8
ˆ 1:30:
Example 11.3. Problem: an experiment on lung tissue elasticity as a function
of lung expansion properties is performed, and the measurements given in

2
Table 11.2 are those of the tissue’s Young’s modulus (Y ), in g cm , at varying

2
f
values of lung expansion in terms of stress (x), in g cm . Assuming that E Y g
2
is linearly related to x and that ˆ 2 (a constant), determine the least-square
Y
estimates of the regression coefficients and an unbiased estimate of 2 .

2

2
Table 11.2 Young’s modulus, y (g cm ), with stress, x (g cm ), for Example 11.3
x 2 2.5 3 5 7 9 10 12 15 16 17 18 19 20
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

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