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

610 CHAPTER 18 | The Binomial Test
■ Score Boundaries and the Binomial Test
In Chapter 6, we noted that a binomial distribution forms a discrete histogram (see
Figure 18.1), whereas the normal distribution is a continuous curve. The difference
between the two distributions was illustrated in Figure 6.18, which is repeated here as
Figure 18.2. In the binomial distribution (the histogram), each score is represented by a
bar (Figure 18.2) and each bar has an upper and a lower boundary. For example, a score
of X = 6 actually corresponds to a bar that reaches from X = 5.5 to X = 6.5. In the
normal approximation to the binomial distribution, each of the bars corresponds to an
interval with upper and lower real limits. For example, in the continuous distribution, a
score of X = 6 corresponds to an interval that extends from a lower real limit of 5.5 to
an upper real limit of 6.5.
When conducting a hypothesis test with the binomial distribution, the basic question
is whether a specific score is located in the critical region. However, because each score
actually corresponds to an interval in the normal distribution, it is possible that part of
the score is in the critical region and part is not. Fortunately, this is usually not an issue.
When pn and qn are both equal to or greater than 10 (the criteria for using the normal
approximation), each interval in the binomial distribution is extremely small and it is
very unlikely that the interval overlaps the critical boundary. For example, the experiment
in Example 18.1 produced a score of X = 3, and we computed a z-score of z = −4.04.
Because this value is in the critical region, beyond z = −1.96, we rejected H 0
. If we had
used the interval boundaries of X = 2.5 and X = 3.5, instead of X = 3, we would have
obtained z-scores of
2.5 2 13.5
z 5
2.60
524.23
3.5 2 13.5
and z 5
2.60
523.85
Note that X 5 3 corresponds
to z 5 4.04,
which is exactly in the
middle of the interval.
Thus, a score of X = 3 actually corresponds to an interval of z-scores ranging from
z = −3.85 to z = −4.23. However, this entire interval is in the critical region beyond
z = −1.96, so the decision is still to reject H 0
.
In most situations, if the whole number X value (in this case, X = 3) is in the critical
region, then the entire interval will also be in the critical region and the correct decision
is to reject H 0
. The only exception to this general rule occurs when an X value produces
a z-score that is in the critical region by a very small margin. In this situation, you should
compute the z-scores corresponding to both ends of the interval to determine whether any
part of the z-score interval is not located in the critical region. Suppose, for example, that
FIGURE 18.2
The relationship between the
binomial distribution and the
normal distribution. The binomial
distribution is always a discrete
histogram, and the normal distribution
is a continuous, smooth curve.
Each X value is represented by a bar
in the histogram or a section of the
normal distribution.
0 1 2 3 4 5 6 7 8 9 10
X