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306 Fundamentals of Probability and Statistics for Engineers
In this approximation, sample mean X is at the center of the interval for
which the width is a function of the sample and the sample size.
Example 9.21. Problem: in a random sample of 500 persons in the city of Los
Angeles it was found that 372 did not approve of US energy policy. Determine
a 95% confidence interval for p, the actual proportion of the Los Angeles
population registering disapproval.
Answer: in this example, n 500, 0.05, and the observed sample mean is
x 372/500 0.74. Table A.3 gives u 0:025 1:96. Substituting these values
into Equation (9.153) then yields
P
0:74 0:04 < p < 0:74 0:04 P
0:70 < p < 0:78 0:95:
REFERENCES
Anderson, R.L., and Bancroft, T.A., 1952, Statistical Theory in Research, McGraw-Hill,
New York.
Benedict, T.R., and Soong, T.T., 1967, ‘‘The Joint Estimation of Signal and Noise from
the Sum Envelope’’, IEEE Trans. Information Theory, IT-13 447–454.
Fisher, R.A., 1922, ‘‘On the Mathematical Foundation of Theoretical Statistics’’, Phil.
Trans. Roy. Soc. London, Ser. A 222 309–368.
Gerlough, D.L., 1955, ‘‘The Use of Poisson Distribution in Traffic’’, in Poisson and
Traffic, The Eno Foundation for Highway Traffic Control, Saugatuk, CT.
Mood, A.M., 1950, Introduction to the Theory of Statistics, McGraw-Hill, New York.
Neyman, J., 1935, ‘‘Su un Teorema Concernente le Cosiddeti Statistiche Sufficienti’’,
Giorn. Inst. Ital. Atturi. 6 320–334.
Pearson, K., 1894, ‘‘Contributions to the Mathematical Theory of Evolution’’, Phil.
Trans. Roy. Soc. London, Ser. A 185 71–78.
Soong, T.T., 1969, ‘‘An Extension of the Moment Method in Statistical Estimation’’,
SIAM J. Appl. Math. 17 560–568.
Wilks, S.S., 1962, Mathematical Statistics, John Wiley & Sons Inc., New York.
FURTHER READING AND COMMENTS
The Crame ´r–Rao inequality is named after two well-known statisticians, H. Crame ´rand
C.R. Rao, who independently established this result in the following references. How-
ever, this inequality was first stated by Fisher in 1922 (see the Reference section). In fact,
much of the foundation of parameter estimation and statistical inference in general, such
as concepts of consistency, efficiency, and sufficiency, was laid down by Fisher in a series
of publications, given below.
Crame ´r, H., 1946, Mathematical Methods of Statistics, Princeton University Press,
Princeton, NJ.
Fisher, R.A., 1924, ‘‘On a Distribution Yielding the Error Functions of Several Well-
known Statistics’’, Proc. Int. Math. Congress, Vol. II, Toronto, 805–813.
TLFeBOOK
