082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

29.14 The normal distribution 969
EXERCISES29.14
1 Arandom variable,x,has a standard normal
distribution. Calculate the probability thatxlies in the
followingintervals:
(a) (0.25, 0.75)
(b) (−0.3,0.1)
(c) within 1.5standard deviationsofthe mean
(d) more thantwo standard deviations from the mean
(e) (−1.7, −0.2)
2 Arandom variable,x,has a normal distribution with
mean 4 andstandard deviation 0.8. Calculate the
probability that
(a) 3.0 x4.4
(b) 2.5 <x<3.9
(c) x > 4.6
(d) x < 4.2
(e) x is within 0.6ofthe mean
3 Arandom variable,t,has anormal distribution with
mean 1 andstandard deviation 2.5. Calculate the
probability that
(a) −1t2 (b)t>0
(c) |t| 0.9 (d) |t| > 1.6
4 Thescores fromIQ tests have amean of100 anda
standard deviation of15. What should aperson score
in order to be described asin the top10% ofthe
population?
5 Amachine produces car pistons.Thediameter ofthe
pistonsfollows anormal distribution, mean 6.04 cm
with astandard deviation of0.02 cm. Thepistonis
acceptable ifitsdiameter is in the range 6.010 cm to
6.055cm. What percentage ofpistonsis
acceptable?
6 The random variable,x, has anormal distribution.
How many standard deviations above the mean must
the pointPbe placed ifthe tail-end isto represent
(a) 10% (b) 5% (c) 1%
ofthe total area? (See Figure29.20.)
N(x)
? P
Figure29.20
Graph forQuestion6.
7 ConsiderFigure29.21. Thetwo tail-endshave equal
area. How many standard deviations from the mean
mustAandBbe placed ifthe tail-ends are
(a) 10% (b) 5% (c) 1%
ofthe total area?
A
N(x)
Figure29.21
Graph forQuestion7.
B
x
x
Solutions
1 (a) 0.1747 (b) 0.1577 (c) 0.8664
(d) 0.0455 (e) 0.3762
2 (a) 0.5858 (b) 0.4199 (c) 0.2266
(d) 0.5987 (e) 0.5467
3 (a) 0.4436 (b) 0.6554 (c) 0.2604
(d) 0.5544
4 119
5 71%
6 (a)1.28 (b)1.645 (c)2.33
7 (a)1.64 (b)1.96 (c)2.57