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

176 CHAPTER 6 | Probability
X
z-score formula
z-score
FIGURE 6.13
Determining probabilities or proportions for a
normal distribution is shown as a two-step process
with z-scores as an intermediate stop along the way.
Note that you cannot move directly along the dashed
line between X values and probabilities and proportions.
Instead, you must follow the solid lines around
the corner.
Unit
normal
table
Proportions
or
probabilities
begin with a specific proportion, use the unit normal table to look up the corresponding z-score,
and then transform the z-score into an X value. The following example demonstrates this process.
EXAMPLE 6.10
The U.S. Census Bureau (2005) reports that Americans spend an average of μ = 24.3 minutes
commuting to work each day. Assuming that the distribution of commuting times is
normal with a standard deviation of σ = 10 minutes, how much time do you have to spend
commuting each day to be in the highest 10% nationwide? (An alternative form of the same
question is presented in Box 6.1.). The distribution is shown in Figure 6.14 with a portion
representing approximately 10% shaded in the right-hand tail.
In this problem, we begin with a proportion (10% or 0.10), and we are looking for a
score. According to the map in Figure 6.13, we can move from p (proportion) to X (score)
via z-scores. The first step is to use the unit normal table to find the z-score that corresponds
to a proportion of 0.10 in the tail. First, scan the values in column C to locate the row that
has a proportion of 0.10 in the tail of the distribution. Note that you will not find 0.1000
exactly, but locate the closest value possible. In this case, the closest value is 0.1003. Reading
across the row, we find z = 1.28 in column A.
The next step is to determine whether the z-score is positive or negative. Remember that the
table does not specify the sign of the z-score. Looking at the distribution in Figure 6.14, you
should realize that the score we want is above the mean, so the z-score is positive, z = +1.28.
Highest 10%
s 5 10
FIGURE 6.14
The distribution of commuting times
for American workers. The problem
is to find the score that separates the
highest 10% of commuting times
from the rest.
m 5 24.3
37.1
0 1.28