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The Essentials of Biostatistics for Physicians, Nurses, and Clinicians,
First Edition. Michael R. Chernick.
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.
CHAPTER 4
Normal Distribution
4. Normal Distribution and Related Properties
4.1 AVERAGES AND THE CENTRAL
LIMIT THEOREM
How does the sample mean behave? If the sample comes from a normal
distribution with mean μ and standard deviation σ , then the sample
average of n observations is also normal with the mean μ , but with
standard deviation σ/ n. So the nice thing here is that the standard
deviation gets smaller as n increases. This means that our estimate (the
sample mean) is an unbiased estimator of μ , and so it tends to get closer
to μ as n gets large.
However, even knowing that we cannot make exact inference
because what we actually know is that ZX=−()/( )ˆ μσ/ n=
nX()ˆ−μσ/ , where X
^
is the sample mean, has a normal distribution
with mean 0 and variance 1. To draw inference about μ we need to
know σ. Because σ causes diffi culties, we call it a nuisance parameter.
In the late nineteenth century and in the fi rst decade of the twentieth
century, researchers would replace σ with a consistent estimate of it,
the sample standard deviation S. They would then do the inference
assuming that nX()Sˆ−μ / has a standard normal distribution.
