ONE SAMPLE TESTS
Procedure & Ho, Ha
Conditions/assumptions
Test Statistic, df if applicable
Large Sample test comparing
univariate mean to a known
or hypothesized value of μ
when the population σ is
known.
Random & Normal (told pop
was normal or large sample
(n ≥ 30, or std norm/box
whisker plot looks normal)
The sample must be <10% of
pop so that (σ/√ ) is valid
for the SD of sample
distribution.
Rand & Norm (told pop was
normal or large sample (n ≥
30, n ≥15 if not to skewed or
std norm/box whisker plot
looks normal). The sample
must be < 10% of the pop
that (σ/√ ) is valid for the
SD of sample dist.
Random & Normal (n ≥ 10 in
each group. Some books say
≥ 5 in each group) & the
sample size is < 10% of
population.
z = ( ̅ -μ)
H
o:μ =Hypo value
H
a:μ ≠ Hypo value
Large sample test comparing
univariate mean to a known
or hypothesized value of μ
when the population σ is
NOT known.
H
y
p
ov
a
l
u
e
o:μ =H
H
y
p
ov
a
l
u
e
a:μ ≠ H
Large Sample test comparing
univariate proportion to a
known or hypothesized value
of p.
H
=H
y
p
ov
a
l
u
e
o:p
H
≠ H
y
p
ov
a
l
u
e
a:p
Confidence Interval
̅ ± crit z (σ/√ )
Calculator Command
Z-Test
(σ/√ )
ZInterval
t = ( ̅ -μ)
̅ ± crit t (s
/√ )
T-Test
(s
/√ )
df = n-1
TInterval
p ±.crit z ((p (1 − p))/n)
1-PropZTest
df = n-1
z =
(p-p)
((p (1 − p))/n)
1-PropZInt
TWO SAMPLE TESTS
Procedure & Ho, Ha
Conditions/assumptions
Large sample indep. test
comparing difference in
bivariate means to a known or
hypothesized difference of μ1
–μ2when the population σ
is known.
Ho: μ1 -μ2=H
y
p
ov
a
l
u
e
Ha: μ1μ2 H
y
p
ov
a
l
u
eo
r
Ho: μ1=
μ2&Ha: μ1≠μ2
Large sample indep test
comparing difference in
bivariate means to a known or
hypothesized difference of μ1
-μ2when the population σ
is NOT known.
Random & Normal
z = ( ̅ 1 - ̅ 2)-( μ1-μ2)
(told pop was normal,
√[(σ21/n1)+(σ22/n2)]
n ≥ 30, or std norm/box
whisker plot looks
normal), &
independent samples
Random, Normal (told
pop was normal, n1&n2
≥ 30, or n1&n2 ≥ 15 if
not to skewed or std
norm/box whisker plot
doesn’t look to
Ho: μ1 -μ2=H
y
p
ov
a
l
u
e skewed), &
Ha: μ1 -μ2≠H
y
p
ov
a
l
u
e independent samples
o
rHo: μ1=
μ2&
Ha: μ1≠μ2
Test Statistic, df if applicable
t = ( ̅ 1 - ̅ 2) -( μ1-μ2)
√[(s21/n1)+(s22/n2)]
Calculator
2-SampZTest
2-SampZInt
̅ 1 - ̅ 2 ± crit t √[(s21/n1)+(s22/n2)]
2-SampTTest
2-SampTInt
df = Smallest group size - 1
(conservative)
or
(V1 + V2)2
df = V12 + V22
n1-1 n2-1
Random & Normal (≥
10 in each group. Some
books say ≥ 5 in each
hypothesized difference of p1 group), independent
samples each
-p2
population is at least
H
y
p
ov
a
l
u
e
o:p1 -p2 =H
10 times as big as its
H
y
p
ov
a
l
u
e
a:p1 -p2 ≠ H
sample
z = (p1- p2)-( p1-p2)
√((( p1(1-p1))/n1)+(( p2(1p2))/n2))
or using pc:
z = (p1- p2)-( p1-p2)
√((( pc(1-pc))/n1)+(( pc(1pc))/n2))
Large sample dependent
test comparing bivariate
mean to a known or
hypothesized value of μd
t = ̅ d - μd
(sd/√n)
Large Sample test
comparing bivariate
proportion to a known or
Confidence Interval
̅ 1 - ̅ 2 ± crit z √[(σ21/n1)+(σ22/n2)]
df = conservative or use formula for
independent t df
(p1- p2) ± crit z √(((p1(1p1))/n1)+(( p2(1-p2))/n2))
2-PropZTest
2-PropZInt
o
rHo: p1=
p2 &Ha: p1≠p2
Ho: μd =H
y
p
ov
a
l
u
e
Ha: μd ≠ H
y
p
ov
a
l
u
e
Random, Normal (told
pop was normal, n ≥
30, or n ≥ 15 & std
norm/box whisker plot
doesn’t look to
skewed), & dependent
samples
df = n-1
̅ d ± crit t (sd/√n)
T-Test
TInterval
df = n-1
Miscellaneous
Procedure & Ho, Ha
c2 goodness of fit
Conditions/assumptions
Random, ALL expected
H
p2…pn=H
y
p
ov
a
l
u
e values must be ≥5 & the
o:p1=
population is at least 10
H
a:H
on
ottr
ue
times as large as the sample
Test Statistic, df if applicable
c2 = Σ(
(obs-exp)2/exp)
Confidence Interval/Other
normal not required
Calculator Command
c2 - GOF-Test
normal not required
c2 -Test
r ± crit t √((1-r2)/(n-2))
LinRegTTest
df = n - 2
LinRegTInt
df = (#col-1)
or
H
y
p
ov
a
l
u
e
o:p1=H
p2=H
y
p
ov
a
l
u
e
…pn=H
y
p
ov
a
l
u
e
H
a:H
on
ottr
ue
c2 Test of homogeneity or
Association
H
i
s
t
r
i
b
u
t
i
o
n
srH
o
m
o
o:D
H
:
D
i
s
t
r
i
b
u
t
i
o
n
s
r
n
o
t
a
H
o
m
o
Random, independent
sampling, ALL expected
values must be ≥5 & the
population is at least 10
times as large as the sample
c2 = Σ(
(obs-exp)2/ exp)
df = (#col-1)(#rows-1)
o
r
H
i
s
t
r
i
b
u
t
i
o
n
srI
n
d
o:D
H
i
s
t
r
i
b
u
t
i
o
n
srn
o
tI
n
d
a:D
Test to see if the regression Random, Linear, Normal (y
(r) or the slope (b) is
pops norm for any given x,
different than zero.
n ≥ 30, or n ≥ 15 & std
norm/box whisker plot
doesn’t look to skewed for
H
=0 o
r H
o:b
o:r
both y data) & y is ind,
=0
H
≠ 0
H
:r resid has no pattern
a:b
a
≠ 0
t=
r
.
√((1-r2)/(n-2))
df = n - 2