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