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

FOCUS ON PROBLEM SOLVING 187
7. The binomial distribution gives the probability
for each value of X, where X equals the number of
occurrences of category A in a series of n events. For
example, X equals the number of heads in n = 10
tosses of a coin.
8. When pn and qn are both at least 10, the binomial
distribution is closely approximated by a normal
distribution with
μ = pn
5 Ïnpq
9. In the normal approximation to the binomial distribution,
each value of X has a corresponding z-score:
z 5 X 2m
s
5 X 2 pn
Ïnpq
With the z-score and the unit normal table, you can
find probability values associated with any value of
X. For maximum accuracy, you should use the appropriate
real limits for X when computing z-scores and
probabilities.
KEY TERMS
probability (161)
random sample (162)
sampling with replacement (163)
unit normal table (168)
percentile rank (178)
percentile (178)
binomial distribution (179)
normal approximation
(binomial) (181)
SPSS ®
The statistics computer package SPSS is not structured to compute probabilities. However, the
program does report probability values as part of the inferential statistics that we will examine
later in this book. In the context of inferential statistics, the probabilities are called significance
levels, and they warn researchers about the probability of misinterpreting their research results.
FOCUS ON PROBLEM SOLVING
1. We have defined probability as being equivalent to a proportion, which means that you
can restate every probability problem as a proportion problem. This definition is particularly
useful when you are working with frequency distribution graphs in which the population
is represented by the whole graph and probabilities (proportions) are represented by
portions of the graph. When working problems with the normal distribution, you always
should start with a sketch of the distribution. You should shade the portion of the graph
that reflects the proportion you are looking for.
2. Remember that the unit normal table shows only positive z-scores in column A. However,
since the normal distribution is symmetrical, the proportions in the table apply to both
positive and negative z-score values.
3. A common error for students is to use negative values for proportions on the left-hand
side of the normal distribution. Proportions (or probabilities) are always positive: 10% is
10% whether it is in the left or right tail of the distribution.
4. The proportions in the unit normal table are accurate only for normal distributions. If a
distribution is not normal, you cannot use the table.
5. For maximum accuracy when using the normal approximation to the binomial distribution,
you must remember that each X value is an interval bounded by real limits. For
example, to find the probability of obtaining an X value greater than 10, you should use
the real limit 10.5 in the z-score formula. Similarly, to find the probability of obtaining an
X value less than 10, you should use the real limit 9.5.