Hypothesis Testing Cheat Sheet
Type
Population Mean
Condition
Difference between
Population Variances
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Notes
H 0 : μ=μ0
H 1 : μ ≠ μ 0 / μ >μ 0 / μ <μ 0
X̄ −μ
Z= σ
√n
Z follows the the standard normal distribution.
σ is known
n is small ( n≤30 )
H 0 : μ=μ0
H 1 : μ ≠ μ 0 / μ >μ 0 / μ <μ 0
X̄− μ
t= σ
√n
t follows the t distribution with n−1 d.o.f.
σ is unknown
n is large ( n>30 )
H 0 : μ=μ0
H 1 : μ ≠ μ 0 / μ >μ 0 / μ <μ 0
Z=
σ is unknown
n is small ( n≤30 )
H 0 : μ=μ0
H 1 : μ ≠ μ 0 / μ >μ 0 / μ <μ 0
t=
H 0 : π =π 0
H 1 : π ≠ π 0 / π >π 0 / π <π 0
Z=
H 0 : π 1−π 2=0
H 1 : π 1−π 2 ≠ 0 / π 1−π 2> 0 / π 1−π 2< 0
Z=
Difference between
Population Proportions
Population Variance
Statistic
99271274
σ is known
n is large ( n>30 )
Population Proportion
Difference between
Population Means
Hypothesis
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σ 1 , σ 2 are known
X̄ −μ
s
√n
Z follows the the standard normal distribution.
X̄− μ
s
√n
t follows the t distribution with n−1 d.o.f.
H 0 : μ1 −μ 2=μ0
H 1 : μ1−μ 2 ≠ μ 0 / μ1−μ2 > μ0 / μ1−μ2 < μ0
Z=
σ 1 , σ 2 are unknown
σ 1 , σ 2 are unequal
n1 , n2 are large
H 0 : μ1 −μ 2=μ0
H 1 : μ1−μ 2 ≠ μ 0 / μ1−μ2 > μ0 / μ1−μ2 < μ0
Z=
σ 1 , σ 2 are unknown
σ 1 , σ 2 are equal
H 0 : μ1 −μ 2=μ0
H 1 : μ1−μ 2 ≠ μ 0 / μ1−μ2 > μ0 / μ1−μ2 < μ0
t=
2
2
H 0 : σ =σ 0
H 1 : σ 2 ≠ σ 20
2
Z follows the the standard normal distribution.
π (1−π )
n
p1− p2−(π 1−π 2)
P(1−P)(
1 1
+ )
n1 n2
x¯1− x¯2−( μ 1−μ 2)
√
√
2
χ 2=
2
F=
2
s2
Z follows the the standard normal distribution.
2
Z follows the the standard normal distribution.
2
s1 s 2
+
n1 n2
S 2p (
1 1
+ )
n1 n2
(n−1)s
σ2
s1
subjects in group 1, 2 resp. that have the characteristic
2
x¯1− x¯2−( μ 1−μ 2)
√
Z follows the the standard normal distribution.
x +x
P= 1 2 where x 1 , x 2 is the number of
n1 +n 2
σ1 σ 2
+
n1 n2
x¯1− x¯2−(μ 1−μ2)
2
H 0 : σ 1= σ 2
2
2
H 1: σ 1≠σ 2
√
√
p−π
2
t follows the t distribution with n1 +n2−2
d.o.f.
χ 2 follows the χ 2 distribution with n−1
d.o.f.
F follows the F distribution with n1−1 and
n2−1 d.o.f.