566 Higher Engineering Mathematics
likely that 1500× 0 .0860, i.e.129 bottles will be
in this range.Now try the following exerciseExercise 215 Further problems on the
introduction to the normal distribution- A component is classed as defective if it has a
diameter of less than 69mm. In a batch of 350
components, the mean diameter is 75mm and
the standard deviation is 2.8mm. Assuming
the diameters are normally distributed,
determine how many are likely to be classed
as defective. [6] - The masses of 800 people are normally dis-
tributed, having a mean value of 64.7kg and a
standard deviation of 5.4kg. Find how many
people are likely to have masses of less than
54.4kg. [22] - 500 tins of paint have a mean content of
1010ml and the standard deviation of the con-
tents is 8.7ml. Assuming the volumes of the
contentsarenormallydistributed,calculatethe
number of tins likely to have contents whose
volumes are less than (a) 1025ml (b) 1000ml
and (c) 995ml.
[(a) 479 (b) 63 (c) 21] - For the 350 components in Problem 1, if those
having a diameter of more than 81.5mm are
rejected, find, correct to the nearest compo-
nent, the number likely to be rejected due to
being oversized. [4] - For the 800 people in Problem 2, determine
how many are likely to have masses of more
than (a) 70kg and (b) 62kg.
[(a) 131 (b) 553] - The mean diameter of holes produced by a
drilling machine bit is 4.05mm and the stan-
dard deviation of the diameters is 0.0028mm.
For twenty holes drilled using this machine,
determine, correct to the nearest whole num-
ber, how many are likely to have diameters
of between (a) 4.048 and 4.0553mm and
(b) 4.052 and 4.056mm, assuming the dia-
meters are normally distributed.
[(a) 15 (b) 4]
7. The intelligence quotients of 400 children
have a mean value of 100 and a standard devi-
ation of 14. Assuming that I.Q.’s are normally
distributed, determine the number of children
likely to have I.Q.’s of between (a) 80 and 90,
(b) 90 and 110 and (c) 110 and 130.
[(a) 65 (b) 209 (c) 89]
8. The mean mass of active material in tablets
produced by a manufacturer is 5.00g and the
standard deviation of the masses is 0.036g. In
a bottlecontaining 100 tablets, find how many
tablets are likelyto have masses of (a) between
4.88 and 4.92g, (b) between 4.92 and 5.04g
and (c) more than 5.04g.
[(a) 1 (b) 85 (c) 13]
58.2 Testing for a normal distribution
It should never be assumed that because data is con-
tinuous it automatically follows that it is normally
distributed. One way of checking that data is normally
distributedis by usingnormal probabilitypaper,often
just calledprobability paper. This is special graph
paper which has linear markings on one axis and per-
centage probability values from 0.01 to 99.99 on the
other axis (see Figs. 58.6 and 58.7). The divisionson the
probabilityaxis are such that astraightlinegraphresults
for normally distributed data when percentage cumu-
lative frequency values are plotted against upper class
boundary values. If the points do not lie in a reasonably
straight line, then the data is not normally distributed.
The method used to test thenormalityof adistributionis
showninProblems5and6.Themeanvalueandstandard
deviation of normally distributed data may be deter-
mined using normal probability paper. For normally dis-
tributed data, the area beneath the standardized normal
curve and az-value of unity (i.e. one standard devia-
tion)may be obtainedfrom Table 58.1. For one standard
deviation, this area is 0.3413, i.e. 34.13%. An area of
±1 standard deviation is symmetrically placed on either
side of thez=0 value, i.e. is symmetrically placed
on either side of the 50% cumulative frequency value.
Thus an area corresponding to±1 standard deviation
extendsfrompercentagecumulativefrequencyvaluesof
( 50 + 34. 13 )%to( 50 − 34. 13 )%, i.e. from 84.13% to
15.87%. For most purposes, these values are taken as
84% and 16%. Thus, when using normal probability