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

170 CHAPTER 6 | Probability
Finding Proportions/Probabilities for Specific z-Score Values For each of the
following examples, we begin with a specific z-score value and then use the unit normal
table to find probabilities or proportions associated with the z-score.
EXAMPLE 6.3A
EXAMPLE 6.3B
EXAMPLE 6.3C
Moving to the left on
the X-axis results in
smaller X values and
smaller z-scores. Thus, a
z-score of 23.00 reflects
a smaller value than a
z-score of 21.
What proportion of the normal distribution corresponds to z-score values greater than
z = 1.00? First, you should sketch the distribution and shade in the area you are trying to
determine. This is shown in Figure 6.8(a). In this case, the shaded portion is the tail of the
distribution beyond z = 1.00. To find this shaded area, you simply look for z = 1.00 in
column A to find the appropriate row in the unit normal table. Then scan across the row to
column C (tail) to find the proportion. Using the table in Appendix B, you should find that
the answer is 0.1587.
You also should notice that this same problem could have been phrased as a probability
question. Specifically, we could have asked, “For a normal distribution, what is
the probability of selecting a z-score value greater than z = +1.00?” Again, the answer is
p(z > 1.00) = 0.1587 (or 15.87%).
■
For a normal distribution, what is the probability of selecting a z-score less than z = 1.50?
In symbols, p(z < 1.50) = ? Our goal is to determine what proportion of the normal
distribution corresponds to z-scores less than 1.50. A normal distribution is shown in
Figure 6.8(b) and z = 1.50 is located in the distribution. Note that we have shaded all the
values to the left of (less than) z = 1.50. This is the portion we are trying to find. Clearly, the
shaded portion is more than 50% so it corresponds to the body of the distribution. Therefore,
find z = 1.50 in column A of the unit normal table and read across the row to obtain
the proportion from column B. The answer is p(z < 1.50) = 0.9332 (or 93.32%). ■
Many problems require that you find proportions for negative z-scores. For example, what
proportion of the normal distribution is contained in the tail beyond z = –0.50? That is,
p(z < –0.50). This portion has been shaded in Figure 6.8(c). To answer questions with
negative z-scores, simply remember that the normal distribution is symmetrical with a
z-score of zero at the mean, positive values to the right, and negative values to the left.
The proportion in the left tail beyond z = –0.50 is identical to the proportion in the right
tail beyond z = +0.50. To find this proportion, look up z = 0.50 in column A, and read
across the row to find the proportion in column C (tail). You should get an answer of
0.3085 (30.85%). ■
The following example is an opportunity for you to test your understanding by finding
proportions in a normal distribution yourself.
(a) (b) (c)
0 1.00
m
FIGURE 6.8
The distributions for Examples 6.3A to 6.3C.
0 1.50
m
20.5 0
m