Higher Engineering Mathematics

560 STATISTICS AND PROBABILITY

mathematical formulae books, and such a table is

shown in Table 58.1, on page 561.

Problem 1. The mean height of 500 people

is 170 cm and the standard deviation is 9 cm.

Assuming the heights are normally distributed,

determine the number of people likely to have

heights between 150 cm and 195 cm.

The mean value,x, is 170 cm and corresponds to
a normal standard variate value,z, of zero on the
standardized normal curve. A height of 150 cm has

az-value given byz=

x−x

σ

standard deviations,

i.e.

150 − 170

9

or−2.22 standard deviations. Using

a table of partial areas beneath the standardized

normal curve (see Table 58.1), az-value of−2.22

corresponds to an area of 0.4868 between the mean

value and the ordinatez=− 2 .22. The negative

z-value shows that it lies to the left of thez= 0

ordinate.

This area is shown shaded in Fig. 58.3(a). Simi-

larly, 195 cm has az-value of

195 − 170

9

that is 2.78

standard deviations. From Table 58.1, this value ofz

corresponds to an area of 0.4973, the positive value

ofzshowing that it lies to the right of thez=0 ordi-

nate. This area is shown shaded in Fig. 58.3(b). The

total area shaded in Figs. 58.3(a) and (b) is shown in

Fig. 58.3(c) and is 0. 4868 + 0 .4973, i.e. 0.9841 of

the total area beneath the curve.

However, the area is directly proportional to prob-

ability. Thus, the probability that a person will have a

height of between 150 and 195 cm is 0.9841. For a

group of 500 people, 500× 0 .9841, i.e.492 peo-

ple are likely to have heights in this range. The

value of 500× 0 .9841 is 492.05, but since answers

based on a normal probability distribution can only

be approximate, results are usually given correct to

the nearest whole number.

Problem 2. For the group of people given in

Problem 1, find the number of people likely to

have heights of less than 165 cm.

A height of 165 cm corresponds to

165 − 170
9
i.e.−0.56 standard deviations.
The area betweenz=0 and z=− 0 .56 (from
Table 58.1) is 0.2123, shown shaded in Fig. 58.4(a).

−2.22 (^0) z-value
(a)
0 2.78z-value
(b)
−2.22 0 2.78z-value
(c)
Figure 58.3
The total area under the standardized normal curve
is unity and since the curve is symmetrical, it follows
that the total area to the left of thez=0 ordinate is
0.5000. Thus the area to the left of thez=− 0. 56
ordinate (‘left’ means ‘less than’, ‘right’ means
‘more than’) is 0. 5000 − 0 .2123, i.e. 0.2877 of the
total area, which is shown shaded in Fig 58.4(b).
The area is directly proportional to probability and
since the total area beneath the standardized normal
curve is unity, the probability of a person’s height
being less than 165 cm is 0.2877. For a group of 500
people, 500× 0 .2877, i.e.144 people are likely to
have heights of less than 165 cm.
Problem 3. For the group of people given in
Problem 1 find how many people are likely to
have heights of more than 194 cm.
194 cm corresponds to az-value of
194 − 170
9
that is,
2.67 standard deviations. From Table 58.1, the area
betweenz=0,z= 2 .67 and the standardized nor-
mal curve is 0.4962, shown shaded in Fig. 58.5(a).