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

SECTION 18.2 | An Example of the Binomial Test 611
the researchers in Example 18.1 found that 8 out of 27 infants in the visual cliff experiment
moved onto the deep side. A score of X = 8 corresponds to
z 5 8 2 13.5
2.60
z 5 2 5.5
2.60
522.12
Because this value is beyond the − 1.96 boundary, it appears that we should reject H 0
.
However, this z-score is only slightly beyond the critical boundary, so it would be wise to
check both ends of the interval. For X = 8, the interval boundaries are 7.5 and 8.5, which
correspond to z-scores of
7.5 2 13.5
z 5
2.60
522.31
8.5 2 13.5
and z 5
2.60
521.92
Thus, a score of X = 8 corresponds to an interval extending from z = −1.92 to z = −2.31.
However, the critical boundary is z = − 1.96, which means that part of the interval (and
part of the score) is not in the critical region for α = .05. Because X = 8 is not completely
beyond the critical boundary, the probability of obtaining X = 8 is greater than α = .05.
Therefore, the correct decision is to fail to reject H 0
.
In general, it is safe to conduct a binomial test using the whole-number value for X.
However, if you obtain a z-score that is only slightly beyond the critical boundary, you also
should compute the z-scores for both ends of the interval. If any part of the z-score interval
is not in the critical region, the correct decision is to fail to reject H 0
.
The following example is an opportunity to test your understanding of the z-score
statistic used in the binomial test.
EXAMPLE 18.2
If the results of the visual cliff study showed that 9 out of 36 infants crawled off the deep
side, what z-score value would be obtained using Equation 18.1? You should find that the
binomial distribution has μ = 1 2 (36) = 18 and σ = Ï_ 1 2+_ 1 2+_36+ = Ï9 = 3. X = 9 corresponds
z = − 9 3 = −3.00.
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IN THE LITERATURE
Reporting the Results of a Binomial Test
Reporting the results of the binomial test typically consists of describing the data and
reporting the z-score value and the probability that the results are due to chance. It is
also helpful to note that a binomial test was used because z-scores are used in other
hypothesis-testing situations (see, for example, Chapter 8). For Example 18.1, the
report might state:
Three out of 27 infants moved to the deep side of the visual cliff. A binomial
test revealed that there is a significant preference for the shallow side of the cliff,
z = −4.04, p < .05.
Once again, p is less than .05. We have rejected the null hypothesis because it is very
unlikely—probability less than 5%—that these results are simply due to chance.