The i th
quartile is at position (i
×
(n +1))/4
Bayes’ Theorem:
P(B|A) = [P(A|B)
×
P(B)]
÷
P(A), where P(A) = P(A|B)
×
P(B) + P(A|B’)
×
P(B’)
The binomial probability of x successes in n trials:
P(x) = n
C x
π x
(1 -
π
)
(n-x)
where n
C x
represents combinations of n things taken x at a time; n
C x
=
( n
− n !
x )!
x !
.
The Poisson probability of x successes in an interval of length t:
λ x
P ( x )
=
λ x e
− λ x !
= e
λ x !
where
λ
= r
×
t, r the rate of occurrence and t the length of the interval
The exponential probability that the first success will not occur until after an interval of length t
0
:
P(t
≥
t
0
) = e
-
λ where
λ
= r
×
t
0
and successes follow a Poisson distribution.
For the normal distribution z
=
( x
−
σ
μ
)
The SamplingDistribution of : The Sampling Distribution of p:
E ( x )
= μ σ x
=
σ n s x
= s n
E(p) =
π σ p
=
π ×
( 1
− n
π
)
Confidence intervals for
μ
: x
± z
C
× σ x
OR x
± t
C
× s x where C is the confidence level, and the t value has n – 1 degrees of freedom
Confidence interval for
π
: p
± z
C
× s p s p
= p
×
( 1
− n p )
To find the minimum necessary sample size: n
=
⎡ ×
⎢⎣ z e
σ ⎤
⎥⎦
2
OR n
= z
2 × π ×
( 1
− e
2
π
)
For hypothesis tests about
μ
: For hypothesis tests about
π
: z
= x
−
σ x
μ
0
OR t
= x
− s x
μ
0 z
= p
σ
− π p
σ p
=
π
0
×
( 1
− π
0
) n