F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Statistical concepts 9
Example 1.3-1
Determine the area under the normal probability curve between z = 1.2
and z = 1.94.
SOLUTION
Required area = (area between z = 0 and z = 1.94) -
(area between z = 0 and z = 1.2)
= 0.4738 (from Table 1.3--3) - 0.3849
= 0.0889
Example 1.3-2
Determine the probability of a set of observations, believed to be normally
distributed, having values that fall below their mean by 1.64 times their
standard deviation.
SOLUTION
We have to determine the area under the normal probability curve
between the ordinates z = - oo and z = -1.64. The area between z =
-1.64 and z = 0 is, by symmetry, equal to the area between z = 0 and z =
+ 1.64 and is 0.4495 from Table 1.3-3.
The area between z = 0 and z = oo is one half the total area between z =
- oo and z = + oo and is therefore equal to 0.5.
Therefore
area between z = -1.64 and z = oo = 0.4495 + 0.5
= 0.9495
The area between z = - oo and z = -1.64 is given by
(area between z = -oo and z = +oo)
-(area between z = -1.64 and z = +oo)
= 1 - 0.9495
= 0.0505 = 0.05
Ans. There is a 5% probability that the value of an observation would fall
below the mean value of all the observations by more than 1.64
their standard deviation.
Example 1.3-3
A set of concrete cube strengths are normally distributed with a mean of 45
N/mm 2 and a standard deviation of 5 N/mm 2 •
(a) Determine the probabilit~ of a random cube having a strength
between 50 and 60 N/mm .
(b) Determine the range in which we would expect these strengths to fall,
with a probability of 99.9%.
SOLUTION
(a) x, the mean = 45 N/mm 2
a, the standard deviation = 5 N/mm 2