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

SECTION 6.2 | Probability and the Normal Distribution 169
(a)
(b)
Body
0.5987
Tail
0.4013
Tail
0.4013
Body
0.5987
z
z
0 10.25
FIGURE 6.7
Proportions of a normal distribution corresponding to z = +0.25 and z = –0.25.
10.25 0
We will use the distribution in Figure 6.7(a) to help introduce the unit normal table. The
figure shows a normal distribution with a vertical line drawn at z = +0.25. Using the portion
of the table shown in Figure 6.6, find the row in the table that contains z = 0.25 in column A.
Reading across the row, you should find that the line drawn z = +0.25 separates the distribution
into two sections with the larger section (the body) containing 0.5987 or 59.87% of the
distribution and the smaller section (the tail) containing 0.4013 or 40.13% of the distribution.
Also, there is exactly 0.0987 or 9.87% of the distribution between the mean and z = +0.25.
To make full use of the unit normal table, there are a few facts to keep in mind:
1. The body always corresponds to the larger part of the distribution whether it is on
the right-hand side or the left-hand side. Similarly, the tail is always the smaller
section whether it is on the right or the left.
2. Because the normal distribution is symmetrical, the proportions on the right-hand
side are exactly the same as the corresponding proportions on the left-hand side.
Earlier, for example, we used the unit normal table to obtain proportions for
z = +0.25. Figure 6.7(b) shows the same proportions for z = –0.25. For a negative
z-score, however, notice that the tail of the distribution is on the left side and
the body is on the right. For a positive z-score (Figure 6.7(a)), the positions are
reversed. However, the proportions in each section are exactly the same, with
0.5987 in the body and 0.4013 in the tail. Once again, the table does not list negative
z-score values. To find proportions for negative z-scores, you must look up the
corresponding proportions for the positive value of z.
3. Although the z-score values change signs (+ and –) from one side to the other,
the proportions are always positive. Thus, column C in the table always lists the
proportion in the tail whether it is the right-hand tail or the left-hand tail.
■ Probabilities, Proportions, and z-Scores
The unit normal table lists relationships between z-score locations and proportions in a normal
distribution. For any z-score location, you can use the table to look up the corresponding
proportions. Similarly, if you know the proportions, you can use the table to find the specific
z-score location. Because we have defined probability as equivalent to proportion, you can
also use the unit normal table to look up probabilities for normal distributions. The following
examples demonstrate a variety of different ways that the unit normal table can be used.