The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

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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 5

Estimating Means

and Proportions

5.1 THE BINOMIAL AND POISSON DISTRIBUTIONS

Consider a discrete variable that has two possible values, such as
success or failure (e.g., success could be complete remission, while
failure would be incomplete or no remission). Let 1 denote success and
0 denote failure. Suppose that we want to determine the proportion of
successes in a population that for practical purposes we can consider
to be infi nite. We take a simple random sample of size n. We can con-
sider this sample to represent a set of observations of n independent
identically distributed random variables that each have probability p to
be a success and 1 − p to be a failure Then the number of successes is
a discrete random variable with parameters n and p , and is called the
binomial distribution. As n gets large, the central limit theorem applies,
and even though the binomial distribution is discrete and the normal
distribution is continuous, the binomial is well approximated by the
normal distribution. Sometimes, to improve the approximation due to
the discrete nature of the binomial, a continuity correction is applied.
However, with the increased speed of the modern computer, it is now
very feasible to do exact inference using the Clopper — Pearson
approach.