Essential formulae 673
Estimating the mean of a population
(σ known):
The confidence coefficient for a large sample size,
(N ≥ 30) is z c where:
Confidence Confidence
level %
coefficient z c
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 ±
z c σ
√
N
N p − N
N p − 1
for a finite population of size N p , and by
x ±
z c σ
√
N
for an infinite population
Estimating the mean of a population
(σ unknown):
The confidence limits of a population mean based on
sample data are given by: μ x ± z c σ x .
Estimating the standard deviation of a
population:
The confidence limits of the standard deviation of a population based on sample data are given by:
s ± z c σ s .
Estimating the mean of a population based
on a small sample size:
The confidence coefficient for a small sample size
(N < 30) is t c which can be determined using Table 74.1,
page 33, on the website. The confidence limits of a
population mean based on sample data is given by:
x ±
t c s
√
(N − 1)
Laplace Transforms
Function
Laplace transforms
f (t )
L{ f (t )} =
∞
0 e −st f (t ) dt
1
1
s
k
k
s
e at
1
s−a
sin at
a
s 2 +a 2
cos at
s
s 2 +a 2
t
1
s 2
t n (n = positve integer)
n!
s n+1
cosh at
s
s 2 −a 2
sinh at
a
s 2 −a 2
e −at t n
n!
(s+a) n+1
e −at sin ωt
ω
(s+a) 2 +ω 2
e −at cos ωt
s+a
(s+a) 2 +ω 2
e −at cosh ωt
s+a
(s+a) 2 −ω 2
e
−at sinh ωt
ω
(s+a) 2 −ω 2