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LESSON 2.5 • Normal Distributions: Finding Areas from Values 133

Figure 2.16(b) shows the standard normal distribution with the area to the right

of z = 1.39 shaded. Again, notice that the shaded areas in the two graphs are the
same.

TEACHING TIP Lesson 2.5

Some students will prefer using a table
of standard normal probabilities/areas
2.19 3.74 5.29 6.84 8.39 9 9.94 11.49 –3 –2 –1 0 1 1.39 2 3 (like Table A) to using technology. In
(a) ITBS vocabulary score (b) z-score
our experience, they tend to be in the
(C) 2021 BFW Publishers -- for review purposes only.
FIGURE 2.16 (a) Normal distribution estimating the proportion of Gary, Indiana, seventh-graders minority, but they love their tables! Expect
who earn ITBS vocabulary scores at the 9th-grade level or higher. (b) The corresponding area in the to have a few of these students in each
standard normal distribution.
class and prepare for class activities
accordingly.
To find the area to the right of = 1.39 locate 1.3 in the left-hand column of z .07 .08 .09

,

z
,
Table A then locate the remaining digit 9 as .09 in the top row. The entry to the right 1.2 .8980 .8997 .9015
of 1.3 and under .09 is .9177. However, this is the area to the left of 1.39=z . We can
use the fact that the total area in the standard normal distribution is 1 to find that 1.3 .9147 .9162 .9177 AL TERNA TE EX AMPLE
.
1

the area to the right of 1.39=z is − 0.9177 = 0.0823 We estimate that about 8.23% 1.4 .9292 .9306 .9319

of Gary, Indiana, seventh-graders earn scores at the ninth-grade level or above on the The greatest on grass?
ITBS vocabulary test.
!
A common student mistake is to look up a z -score in Table A and report the entry caution Finding area to the right in a normal
aution

corresponding to that z -score, regardless of whether the problem asks for the area to
the left or to the right of that z -score. This mistake can usually be prevented by follow- distribution
ing the two-step process, which includes drawing a normal distribution and shading PROBLEM: In 2017, Roger Federer won
the area of interest. Look at the sketch to see if the area should be closer to 0 or closer

to 1. In the previous scenario, for instance, it should be obvious that 0.9177 is not the the Wimbledon Championship—one of
area of the shaded region. the most prestigious tournaments in all
of tennis. Wimbledon is played on grass
courts—one of 4 basic types of tennis
EXAMPLE surfaces. This was a record-setting
eighth Wimbledon victory for Federer,
Can Rory clear the trees? and he also became the oldest man
Finding area to the right in a normal distribution to win the Wimbledon Championship.
During this tournament, Federer’s first
PROBLEM: When professional golfer Rory McIlroy hits
his driver, the distance the ball travels can be modeled by serve speeds averaged 115 mph. Assume
a normal distribution with mean 304 yards and standard Maddie Meyer/Getty Images that the distribution of first serve speeds
deviation 8 yards. On a specific hole, Rory would need to is approximately normal with a standard
hit the ball at least 290 yards to have a clear second shot
that avoids a large group of trees. What percent of Rory’s deviation of 4 mph. About what percent
drives travel at least 290 yards? Is he likely to have a clear of Federer’s first serves were at least
second shot?
125 mph? Show your work.
SOLUTION:
Area = 0.0062
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103 107 111 115 119 123 127
125
First serve speed (mph)
125 −115
z = = 2.50
4
(i) Using Table A: Area for
=
z > 2.500.0062
Using technology:
Applet/normalcdf(lower:2.50,
=
upper:1000,mean: 0,SD:1)0.0062
(ii) Applet/normalcdf(lower:125,
upper:1000,mean:115,SD: 4) 0.0062
=
About 0.62% of Federer’s first serves
were faster than 125 mph.
LESSON 2.5 • Normal Distributions: Finding Areas from Values 133

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