190 Some Special Distributionsits pdf is displayed in Figure 3.4.2. Common notation for the cdf ofZisP(Z≤z)=Φ(z)=dfn∫z01
√
2 πe−t(^2) / 2
dt, −∞<z<∞. (3.4.9)
Table II of Appendix D displays a table for Φ(z) for specified values ofz>0. To
compute Φ(−z), wherez>0, use the identity
Φ(−z)=1−Φ(z). (3.4.10)
This identity follows because the pdf ofZis symmetric about 0. It is apparent in
Figure 3.4.2 and the reader is asked to show it in Exercise 3.4.1.
φ(z)
z
zp (0,0)
Φ(zp)=p
Figure 3.4.2:The standard normal density:p=Φ(zp) is the area under the curve
to the left ofzp.
As an illustration of the use of Table II, suppose in Example 3.4.1 that we want
to determine the probability that the height of an adult male is between 67 and 71
inches. This is calculated as
P(67<X<71) = P(X<71)−P(X<67)
= P
(
X− 70
4
<
71 − 70
4
)
−P
(
X− 70
4
<
67 − 70
4
)
= P(Z< 0 .25)−P(Z<− 0 .75) = Φ(0.25)−1+Φ(0.75)
=0. 5987 −1+0.7734 = 0. 3721 (3.4.11)
= pnorm(71, 70 ,4)−pnorm(67, 70 ,4) = 0. 372079. (3.4.12)
Expression (3.4.11) is the calculation by using Table II, while the last line is the cal-
culation by using the R functionpnorm. More examples are offered in the exercises.
As a final note on Table II, it is generated by the R function: