Fundamentals of Probability and Statistics for Engineers

determine the estimated regression line for Y as a function of log 10 v.

11.8 An experimental study of nasal deposition of particles was carried out and
showed a linear relationship between E Y and ln d^2 f , where Y is the fraction
of particles of aerodynamic diameter, d (in mm), that is deposited in the nose
during an inhalation of f(l min^1 ). Consider the data given in Table 11.8 (four
readings are taken at each value of ln d^2 f). Estimate the regression parameters in
the linear regression equation

and estimate^2 , the variance of Y.

11.9 For a study of the stress–strain history of soft biological tissues, experimental
results relating dynamic moduli of aorta (D) to stress fr equency ( ) are given in
Table 11.9.
(a) Assuming that E D , and , estimate regression coefficients
and.
(b) Determine a one-sided 95% confiden ce interval for the variance of D.
(c) Test if the slope estimate is significantly different from zero at the 5%
significance level.

11.10 G iven the data in Table 11.10
(a) D etermine the least-square estimates of ,and 2 assuming that

Table 11. 7 Noise level, y (in dB) with vehicle

speed, v (in km h^1 ), for Problem 11.7

v 2030405060708090100

y 5 563687072787476 79

Table 11. 8 Fraction of particles inhaled of diameter d (in mm), with ln d^2 f

(f is inhalation, in min^1 ), for Problem 11.8

ln d^2 f 1.6 1.7 2.0 2.8 3.0 3.0 3.6

y 0.39 0.41 0.42 0.61 0.83 0.79 0.98

0.30 0.28 0.34 0.51 0.79 0.69 0.88

0.21 0.20 0.22 0.47 0.70 0.63 0.87

0.12 0.10 0.18 0.39 0.61 0.59 0.83

Table 11.9 The dynamic modulus of aorta, d (normalized) with frequency,

(in Hz), for Problem 11.9

12345678910

1.60 1.51 1.40 1.57 1.60 1.59 1.80 1.59 1.82 1.59

Linear Models and Linear Regression 361

f g



EfYgˆ ‡ lnd^2 f;



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f gˆ ‡! ^2 Dˆ^2 ,

0 , 1 ,

EfYgˆ 0 ‡ 1 x 1 ‡ 2 x 2 :



l 

!

!

d