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4. Miller, I., and J.E. Freund, Probability and Statistics for Engineers, 3rd ed., Prentice Hall,
Englewood Cliffs, NJ, 1985.
5. Bury, K., Statistical Distributions in Engineering, Cambridge Press, Cambridge, UK, 1999.
6. Vardeman, S.B., Statistics for Engineering Problem Solving, PWS Publishing Company,
Boston, 1994.
7. ASME/ANSI Power Test Codes, Test Uncertainty PTC 19.1-2005, American Society of
Mechanical Engineers, New York, 2005.
8. Joint Committee for Guides in Metrology (JCGM), Propagation of Distributions Using the
Monte Carlo Method, JCGM 101:2008, JCGM, Sevres Cedex, France, 2008.
9. Devroye, L., Non-Uniform Random Variate Generation, Springer, 1986. See also http://cg.scs.
carleton.ca/$luc/rnbookindex.html.
NOMENCLATURE
a 0 , a 1 , . . . , a m polynomial regression coefficients
f j
frequency of occurrence of a value
x
measured variable; measurand
x i
ith measured value in a data set
x
0
true mean value of the population
of x
x
sample mean value of x
hxi
pooled sample mean of x
p(x)
probability density function of x
s x
sample standard deviation x
s x
sample standard deviation of the
means of x; standard random
uncertainty in x
s
2
x
sample variance of x
hs x i
pooled sample standard deviation
of x
hs x i
pooled standard deviation of the
means of x
hs x i
2
pooled sample variance of x
s yx
standard error of the (curve) fit
between y and x; standard random uncertainty in a curve fit
t
Student’s t variable
b
normalized standard variate
s
true standard deviation of the population of x
s
2
true variance of the population of x
x
2
chi-squared value
n
degrees of freedom
PROBLEMS
4.1 Determine the range of values containing 50% of the population of x. From a large sampling
(N > 5000), we find that x has a mean value of 5.2 units and a standard deviation of 1.0 units. Assume
x is normally distributed.
4.2 Determine the range of values containing 90% of the population of x. From a very large data set (N >
50,000), x is found to have a mean value of 192.0 units with a standard deviation of 10 units. Assume
x is normally distributed.
4.3 At a fixed operating setting, the pressure in a line downstream of a reciprocating compressor has a
mean value of 12.0 bar with a standard deviation of 1.0 bar based on a very large data set obtained
from continuous monitoring. What is the probability that the line pressure will exceed 13.0 bar
during any measurement?
Problems 153
14:7:49 Page 153
4. Miller, I., and J.E. Freund, Probability and Statistics for Engineers, 3rd ed., Prentice Hall,
Englewood Cliffs, NJ, 1985.
5. Bury, K., Statistical Distributions in Engineering, Cambridge Press, Cambridge, UK, 1999.
6. Vardeman, S.B., Statistics for Engineering Problem Solving, PWS Publishing Company,
Boston, 1994.
7. ASME/ANSI Power Test Codes, Test Uncertainty PTC 19.1-2005, American Society of
Mechanical Engineers, New York, 2005.
8. Joint Committee for Guides in Metrology (JCGM), Propagation of Distributions Using the
Monte Carlo Method, JCGM 101:2008, JCGM, Sevres Cedex, France, 2008.
9. Devroye, L., Non-Uniform Random Variate Generation, Springer, 1986. See also http://cg.scs.
carleton.ca/$luc/rnbookindex.html.
NOMENCLATURE
a 0 , a 1 , . . . , a m polynomial regression coefficients
f j
frequency of occurrence of a value
x
measured variable; measurand
x i
ith measured value in a data set
x
0
true mean value of the population
of x
x
sample mean value of x
hxi
pooled sample mean of x
p(x)
probability density function of x
s x
sample standard deviation x
s x
sample standard deviation of the
means of x; standard random
uncertainty in x
s
2
x
sample variance of x
hs x i
pooled sample standard deviation
of x
hs x i
pooled standard deviation of the
means of x
hs x i
2
pooled sample variance of x
s yx
standard error of the (curve) fit
between y and x; standard random uncertainty in a curve fit
t
Student’s t variable
b
normalized standard variate
s
true standard deviation of the population of x
s
2
true variance of the population of x
x
2
chi-squared value
n
degrees of freedom
PROBLEMS
4.1 Determine the range of values containing 50% of the population of x. From a large sampling
(N > 5000), we find that x has a mean value of 5.2 units and a standard deviation of 1.0 units. Assume
x is normally distributed.
4.2 Determine the range of values containing 90% of the population of x. From a very large data set (N >
50,000), x is found to have a mean value of 192.0 units with a standard deviation of 10 units. Assume
x is normally distributed.
4.3 At a fixed operating setting, the pressure in a line downstream of a reciprocating compressor has a
mean value of 12.0 bar with a standard deviation of 1.0 bar based on a very large data set obtained
from continuous monitoring. What is the probability that the line pressure will exceed 13.0 bar
during any measurement?
Problems 153