Higher Engineering Mathematics, Sixth Edition

Essential formulae 673

on a small sample size 74.5 Estimating the mean of a population based

(σknown):

The confidence coefficient for a large sample size,
(N≥ 30 )iszcwhere:

Confidence Confidence

level % coefficientzc

99 2.58

98 2.33

96 2.05

95 1.96

90 1.645

80 1.28

50 0.6745

The confidence limits of a population mean based on
sample data are given by:

x±

zcσ

√

N

√(

Np−N

Np− 1

)

for a finite population of sizeNp, and by

x±

zcσ

√

N

for an infinite population

Estimating the mean of a population

(σunknown):

The confidence limits of a population mean based on
sample data are given by:μx±zcσx.

Estimating the standard deviation of a

population:

The confidence limitsof the standard deviationof a pop-
ulation based on sample data are given by:
s±zcσs.

Estimating the mean of a population based

on a small sample size:

The confidence coefficient for a small sample size
(N< 30 )istcwhichcanbedeterminedusingTable74.1,

page 33, on the website. The confidence limits of a

population mean based on sample data is given by:

x±

tcs

√

(N− 1 )

Laplace Transforms

Function Laplace transforms

f(t) L{f(t)}=

∫∞

0 e

−stf(t)dt

(^11) s
k ks
eat s−^1 a
sinat s (^2) +aa 2
cosat s (^2) +sa 2
t s^12
tn(n=positve integer) snn+! 1
coshat s (^2) −sa 2
sinhat s (^2) −aa 2
e−attn (s+na!)n+ 1
e−atsinωt (s+aω) (^2) +ω 2
e−atcosωt (s+as+) 2 a+ω 2
e−atcoshωt (s+as+) 2 a−ω 2
e−atsinhωt (s+aω) (^2) −ω 2