Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 6.4 | Probability and the Binomial Distribution 179
6.4 Probability and the Binomial Distribution
LEARNING OBJECTIVES
5. Describe the circumstances in which the normal distribution serves as a good
approximation to the binomial distribution.
6. Calculate the probability associated with a specific X value in a binomial distribution
and find the score associated with a specific proportion of a binomial distribution.
When a variable is measured on a scale consisting of exactly two categories, the resulting
data are called binomial. The term binomial can be loosely translated as “two names,”
referring to the two categories on the measurement scale.
Binomial data can occur when a variable naturally exists with only two categories. For
example, people can be classified as male or female, and a coin toss results in either heads
or tails. It also is common for a researcher to simplify data by collapsing the scores into
two categories. For example, a psychologist may use personality scores to classify people
as either high or low in aggression.
In binomial situations, the researcher often knows the probabilities associated with each
of the two categories. With a balanced coin, for example, p(heads) = p(tails) = 1 2 . The
question of interest is the number of times each category occurs in a series of trials or in a
sample of individuals. For example:
What is the probability of obtaining 15 heads in 20 tosses of a balanced coin?
What is the probability of obtaining more than 40 introverts in a sampling of 50 college
freshmen?
As we shall see, the normal distribution serves as an excellent model for computing
probabilities with binomial data.
■ The Binomial Distribution
To answer probability questions about binomial data, we need to examine the binomial
distribution. To define and describe this distribution, we first introduce some notation.
1. The two categories are identified as A and B.
2. The probabilities (or proportions) associated with each category are identified as
p = p(A) = the probability of A
q = p(B) = the probability of B
Notice that p + q = 1.00 because A and B are the only two possible outcomes.
3. The number of individuals or observations in the sample is identified by n.
4. The variable X refers to the number of times category A occurs in the sample.
Notice that X can have any value from 0 (none of the sample is in category A) to n (all
the sample is in category A).
DEFINITION
Using the notation presented here, the binomial distribution shows the probability
associated with each value of X from X = 0 to X = n.
A simple example of a binomial distribution is presented next.