Formulas
Use when σ is known. Single population under study. Single z- test. Single confidence interval.
H 0 : µ = µ0
x -µ
z =
σ
0
x±z
*
 zσ 
n=

 m
σ
n
2
C
Z*
90%
1.645
95%
1.96
99%
2.576
n
Use when σ is known. Two populations under study. Two sample z-test. Two sample z confidence interval.
H 0 : µ1 = µ 2
z=
x1 − x 2
σ 12
n1
+
( x1 − x 2 ) ± z *
σ 22
σ 12
n1
+
σ 22
n2
n2
Use when σ is not known. Single Population under study; t-test. Single t confidence interval.
H 0 : µ = µ0
t =
x - µ0
s
n
x ± t*
s
n
With n - 1 degrees of freedom (df)
Use when σ is not known. Two populations under study. Two sample t –test. Two sample t confidence
interval.
H 0 : µ1 = µ 2
t=
x1 − x 2
s12 s 22
+
n1 n 2
( x1 − x 2 ) ± t
*
s12 s22
+
n1 n 2
With the lesser of n1 -1 or n2 -1 for the
degrees of freedom (df).
Pooled Formulas
s2 p =
( n1 − 1)s12 + ( n 2 − 1)s 22
n1 + n 2 − 2
Two sample t-test pooled s.
H 0 : µ1 = µ 2
t=
x1 − x 2
1
1
sp
+
n1 n 2
Estimating σ, by pooling the standard deviations from
both populations.
Two sample t confidence interval pooled s.
( x 1 − x 2 ) ± t * (s p )
1
1
+
n1 n 2
d.f. = n1 + n2 - 2
Proportions
Single Proportion significance test.
H 0 : p = p0
p̂ =
X
n
p̂ − p o
p o (1 − p o )
n
z =
Wilson Estimates. Single proportion Wilson confidence interval.
X+2
~
p=
n+4
~
p ± z*
~
p (1 − ~
p)
n+4
 z* 

n + 4 = 

 2m 
2
2
 z*  *
n + 4 =   p (1 − p * )
m
Two proportions under study. Two sample proportion significance test.
H 0 : p2 = p1
z=
p̂ 2 - p̂1
1
1 
p̂(1 - p̂) + 
 n1 n 2 
,where pˆ =
X1 + X 2
n1 + n 2
Two sample proportion Wilson confidence interval.
~
p (1 − ~
p (1 − ~
p2 ) ~
p1 )
~
p2 - ~
p1 ± z* 2
+ 1
n2 + 2
n1 + 2
X +1
were ~
p1 = 1
n1 + 2
X +1
~
p2 = 2
n2 + 2