University Foundation Mathematics and Statistics

University
Foundation Mathematics and Statistics
By
Your Name
Date
1

Table of Contents
Executive Summary.........................................................................................3
Introduction.....................................................................................................3
Background......................................................................................................3
Binomial distribution.....................................................................................4
Normal Distribution.......................................................................................4
Central limit theorem....................................................................................5
Why we use normal distribution as an approximation for binomial distribution
.........................................................................................................................5
When we use normal distribution as an approximation for binomial
distribution.......................................................................................................6
How the Technique is Applied..........................................................................6
Procedure......................................................................................................6
Examples....................................................................................................10
Conclusion.....................................................................................................13
References.....................................................................................................14
2

Using the Normal Distribution as an Approximation for the Binomial
Distribution
Executive Summary
Application of normal distribution as an approximation for the binomial
distribution is a technique that is applied when the need to use a continuous
distribution to approximate a discrete distribution arises. In most cases, the
technique is applied to reduce the complexity that arises when using the
binomial formula in scenarios where the random binomial variables are
many. The central limit theorem facilitates the approximation process. The
theorem requires that as sample size increases the sampling distribution of
the sample means become almost normal. Before the approximation is done,
the sample is accessed whether its large enough for the approximation to be
undertaken. If the sample meets the criteria, the binomial distribution in
question is approximated using normal distribution. A set of rules and
procedures is followed to ensure accurate results are obtained for various
probabilities conducted for a particular test. A continuity correction factor is
used to increase the accuracy of the approximation process.
Introduction
The process of computing binomial probabilities become complex as the
number of variables increase for a particular test. To bypass the complexity
that maybe incurred an approximation is done through the utilization of
normal distribution (Hinton, 2014). The approximation process requires
certain criteria and procedures be observed to obtain the utmost optimum
results (Shao, 2010). This paper provides a guide that can be used in using
the normal distribution as an approximation for the binomial distribution. It
presents a background for each variable, why the approximation process is
used, when it can be used and how it can be effectively be used to obtain
accurate inferences for various applicable statistical tests.
3

Background
Normal distribution for approximation to binomial distribution is a technique
that is applied when the need to use a continuous distribution to
approximate a discrete distribution arises. The continuous distribution is
known also known as the normal distribution while the discrete distribution is
known the binomial distribution (Rumsey, 2015). This technique is aided by
the sampling limit theorem which states that that when the sample size is
made large enough, then the sampling distribution of the sample means
become almost normal (Selvanathan and Keller, 2017).
Binomial distribution
Binomial distribution can be described as probability of having outcomes that
can either a success or a failure in an experimental survey or research study
that has been conducted numerous numbers of times. Alternatively, it can be
said that a binomial distribution is a distribution that has got only two
possible outcomes (Lock, 2013). For example, when a coin is tossed, there is
only a possibility of obtaining a head or tail. In binomial distribution, random
variables are of the type discrete. Therefore, only countable number of
observations occur in this type of distribution with a significant separation
between the observations. In other words, it can be said that a discrete
variable in a binomial distribution can take specific values and not any other
values in between. For example, it can take the value five or six and not any
value between five and six. The distribution has to main variables labelled as
n and p. The variable n represents the number of times the study is
undertaken while the variable p represents the probability of a particular
outcome (Linoff, 2011). For example, if a coin is tossed the probability of
success or failure is 0.5 and is the coin is tossed for 20 times, then the
binomial distribution would be (n=20, p=0.5). For a distribution to be
considered as binomial, the following criteria must be observed:
4