<s>
A	O
Z-test	B-General_Concept
is	O
any	O
statistical	O
test	O
for	O
which	O
the	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
can	O
be	O
approximated	O
by	O
a	O
normal	O
distribution	O
.	O
</s>
<s>
Z-tests	B-General_Concept
test	O
the	O
mean	O
of	O
a	O
distribution	O
.	O
</s>
<s>
For	O
each	O
significance	B-General_Concept
level	I-General_Concept
in	O
the	O
confidence	O
interval	O
,	O
the	O
Z-test	B-General_Concept
has	O
a	O
single	O
critical	O
value	O
(	O
for	O
example	O
,	O
1.96	O
for	O
5%	O
two	O
tailed	O
)	O
which	O
makes	O
it	O
more	O
convenient	O
than	O
the	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
whose	O
critical	O
values	O
are	O
defined	O
by	O
the	O
sample	O
size	O
(	O
through	O
the	O
corresponding	O
degrees	O
of	O
freedom	O
)	O
.	O
</s>
<s>
Both	O
the	O
Z-test	B-General_Concept
and	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
have	O
similarities	O
in	O
that	O
they	O
both	O
help	O
determine	O
the	O
significance	O
of	O
a	O
set	O
of	O
data	O
.	O
</s>
<s>
However	O
,	O
the	O
z-test	B-General_Concept
is	O
rarely	O
used	O
in	O
practice	O
because	O
the	O
population	O
deviation	O
is	O
difficult	O
to	O
determine	O
.	O
</s>
<s>
Because	O
of	O
the	O
central	O
limit	O
theorem	O
,	O
many	O
test	B-General_Concept
statistics	I-General_Concept
are	O
approximately	O
normally	O
distributed	O
for	O
large	O
samples	O
.	O
</s>
<s>
Therefore	O
,	O
many	O
statistical	O
tests	O
can	O
be	O
conveniently	O
performed	O
as	O
approximate	O
Z-tests	B-General_Concept
if	O
the	O
sample	O
size	O
is	O
large	O
or	O
the	O
population	O
variance	O
is	O
known	O
.	O
</s>
<s>
If	O
the	O
population	O
variance	O
is	O
unknown	O
(	O
and	O
therefore	O
has	O
to	O
be	O
estimated	O
from	O
the	O
sample	O
itself	O
)	O
and	O
the	O
sample	O
size	O
is	O
not	O
large	O
(	O
n	O
<	O
30	O
)	O
,	O
the	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
may	O
be	O
more	O
appropriate	O
.	O
</s>
<s>
How	O
to	O
perform	O
a	O
Z	B-General_Concept
test	I-General_Concept
when	O
T	O
is	O
a	O
statistic	O
that	O
is	O
approximately	O
normally	O
distributed	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
as	O
follows	O
:	O
</s>
<s>
First	O
,	O
estimate	O
the	O
expected	O
value	O
μ	O
of	O
T	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
and	O
obtain	O
an	O
estimate	O
s	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
T	O
.	O
</s>
<s>
For	O
Null	B-General_Concept
hypothesis	I-General_Concept
H0	O
:	O
μ≥μ0	O
vs	O
alternative	B-General_Concept
hypothesis	I-General_Concept
H1	O
:	O
μ	O
<	O
μ0	O
,	O
it	O
is	O
lower/left	O
-tailed	O
(	O
one	O
tailed	O
)	O
.	O
</s>
<s>
For	O
Null	B-General_Concept
hypothesis	I-General_Concept
H0	O
:	O
μ≤μ0	O
vs	O
alternative	B-General_Concept
hypothesis	I-General_Concept
H1	O
:	O
μ>μ0	O
,	O
it	O
is	O
upper/right	O
-tailed	O
(	O
one	O
tailed	O
)	O
.	O
</s>
<s>
For	O
Null	B-General_Concept
hypothesis	I-General_Concept
H0	O
:	O
μ	O
=	O
μ0	O
vs	O
alternative	B-General_Concept
hypothesis	I-General_Concept
H1	O
:	O
μ≠μ0	O
,	O
it	O
is	O
two-tailed	O
.	O
</s>
<s>
Third	O
,	O
calculate	O
the	O
standard	O
score	O
:	O
which	O
one-tailed	B-General_Concept
and	I-General_Concept
two-tailed	I-General_Concept
p-values	B-General_Concept
can	O
be	O
calculated	O
as	O
Φ(Z )	O
(	O
for	O
lower/left	O
-tailed	O
tests	O
)	O
,	O
Φ(Z )	O
(	O
for	O
upper/right	O
-tailed	O
tests	O
)	O
and	O
2Φ( |Z|	O
)	O
(	O
for	O
two-tailed	B-General_Concept
tests	I-General_Concept
)	O
where	O
Φ	O
is	O
the	O
standard	O
normal	O
cumulative	O
distribution	O
function	O
.	O
</s>
<s>
The	O
term	O
"	O
Z-test	B-General_Concept
"	O
is	O
often	O
used	O
to	O
refer	O
specifically	O
to	O
the	O
one-sample	B-General_Concept
location	I-General_Concept
test	I-General_Concept
comparing	O
the	O
mean	O
of	O
a	O
set	O
of	O
measurements	O
to	O
a	O
given	O
constant	O
when	O
the	O
sample	O
variance	O
is	O
known	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
the	O
mean	O
value	O
of	O
X	O
is	O
a	O
given	O
number	O
μ0	O
.	O
</s>
<s>
We	O
can	O
use	O
X	O
as	O
a	O
test-statistic	O
,	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
X−	O
μ0	O
is	O
large	O
.	O
</s>
<s>
The	O
resulting	O
test	O
will	O
not	O
be	O
an	O
exact	O
Z-test	B-General_Concept
since	O
the	O
uncertainty	O
in	O
the	O
sample	O
variance	O
is	O
not	O
accounted	O
for	O
—	O
however	O
,	O
it	O
will	O
be	O
a	O
good	O
approximation	O
unless	O
the	O
sample	O
size	O
is	O
small	O
.	O
</s>
<s>
A	O
t-test	B-General_Concept
can	O
be	O
used	O
to	O
account	O
for	O
the	O
uncertainty	O
in	O
the	O
sample	O
variance	O
when	O
the	O
data	O
are	O
exactly	O
normal	O
.	O
</s>
<s>
Difference	O
between	O
Z-test	B-General_Concept
and	O
t-test	B-General_Concept
:	O
Z-test	B-General_Concept
is	O
used	O
when	O
sample	O
size	O
is	O
large	O
(	O
n>50	O
)	O
,	O
or	O
the	O
population	O
variance	O
is	O
known	O
.	O
</s>
<s>
t-test	B-General_Concept
is	O
used	O
when	O
sample	O
size	O
is	O
small	O
(	O
n	O
<	O
50	O
)	O
and	O
population	O
variance	O
is	O
unknown	O
.	O
</s>
<s>
For	O
large	O
sample	O
sizes	O
,	O
the	O
t-test	B-General_Concept
procedure	O
gives	O
almost	O
identical	O
p-values	B-General_Concept
as	O
the	O
Z-test	B-General_Concept
procedure	O
.	O
</s>
<s>
Other	O
location	B-General_Concept
tests	I-General_Concept
that	O
can	O
be	O
performed	O
as	O
Z-tests	B-General_Concept
are	O
the	O
two-sample	O
location	B-General_Concept
test	I-General_Concept
and	O
the	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
For	O
the	O
Z-test	B-General_Concept
to	O
be	O
applicable	O
,	O
certain	O
conditions	O
must	O
be	O
met	O
.	O
</s>
<s>
Nuisance	O
parameters	O
should	O
be	O
known	O
,	O
or	O
estimated	O
with	O
high	O
accuracy	O
(	O
an	O
example	O
of	O
a	O
nuisance	O
parameter	O
would	O
be	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
in	O
a	O
one-sample	B-General_Concept
location	I-General_Concept
test	I-General_Concept
)	O
.	O
</s>
<s>
Z-tests	B-General_Concept
focus	O
on	O
a	O
single	O
parameter	O
,	O
and	O
treat	O
all	O
other	O
unknown	O
parameters	O
as	O
being	O
fixed	O
at	O
their	O
true	O
values	O
.	O
</s>
<s>
However	O
if	O
the	O
sample	O
size	O
is	O
not	O
large	O
enough	O
for	O
these	O
estimates	O
to	O
be	O
reasonably	O
accurate	O
,	O
the	O
Z-test	B-General_Concept
may	O
not	O
perform	O
well	O
.	O
</s>
<s>
The	O
test	B-General_Concept
statistic	I-General_Concept
should	O
follow	O
a	O
normal	O
distribution	O
.	O
</s>
<s>
Generally	O
,	O
one	O
appeals	O
to	O
the	O
central	O
limit	O
theorem	O
to	O
justify	O
assuming	O
that	O
a	O
test	B-General_Concept
statistic	I-General_Concept
varies	O
normally	O
.	O
</s>
<s>
There	O
is	O
a	O
great	O
deal	O
of	O
statistical	O
research	O
on	O
the	O
question	O
of	O
when	O
a	O
test	B-General_Concept
statistic	I-General_Concept
varies	O
approximately	O
normally	O
.	O
</s>
<s>
If	O
the	O
variation	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
strongly	O
non-normal	O
,	O
a	O
Z-test	B-General_Concept
should	O
not	O
be	O
used	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
of	O
Z-tests	B-General_Concept
for	O
the	O
one	O
or	O
two	O
sample	O
location	O
problem	O
,	O
the	O
usual	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
only	O
appropriate	O
if	O
the	O
data	O
were	O
collected	O
as	O
an	O
independent	O
sample	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
one	O
and	O
two	O
sample	O
location	O
problems	O
,	O
a	O
t-test	B-General_Concept
does	O
this	O
.	O
</s>
<s>
Suppose	O
that	O
in	O
a	O
particular	O
geographic	O
region	O
,	O
the	O
mean	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
scores	O
on	O
a	O
reading	O
test	O
are	O
100	O
points	O
,	O
and	O
12	O
points	O
,	O
respectively	O
.	O
</s>
<s>
First	O
calculate	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
:	O
</s>
<s>
where	O
is	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Next	O
calculate	O
the	O
z-score	O
,	O
which	O
is	O
the	O
distance	O
from	O
the	O
sample	O
mean	O
to	O
the	O
population	O
mean	O
in	O
units	O
of	O
the	O
standard	B-General_Concept
error	I-General_Concept
:	O
</s>
<s>
When	O
population	O
parameters	O
are	O
unknown	O
,	O
a	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
should	O
be	O
conducted	O
instead	O
.	O
</s>
<s>
The	O
classroom	O
mean	O
score	O
is	O
96	O
,	O
which	O
is	O
−	O
2.47	O
standard	B-General_Concept
error	I-General_Concept
units	O
from	O
the	O
population	O
mean	O
of	O
100	O
.	O
</s>
<s>
This	O
is	O
the	O
one-sided	B-General_Concept
p-value	B-General_Concept
for	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
55	O
students	O
are	O
comparable	O
to	O
a	O
simple	O
random	O
sample	O
from	O
the	O
population	O
of	O
all	O
test-takers	O
.	O
</s>
<s>
The	O
two-sided	O
p-value	B-General_Concept
is	O
approximately	O
0.014	O
(	O
twice	O
the	O
one-sided	B-General_Concept
p-value	B-General_Concept
)	O
.	O
</s>
<s>
We	O
could	O
also	O
say	O
that	O
with	O
98.6	O
%	O
confidence	O
we	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
55	O
test	O
takers	O
are	O
comparable	O
to	O
a	O
simple	O
random	O
sample	O
from	O
the	O
population	O
of	O
test-takers	O
.	O
</s>
<s>
The	O
Z-test	B-General_Concept
tells	O
us	O
that	O
the	O
55	O
students	O
of	O
interest	O
have	O
an	O
unusually	O
low	O
mean	O
test	O
score	O
compared	O
to	O
most	O
simple	O
random	O
samples	O
of	O
similar	O
size	O
from	O
the	O
population	O
of	O
test-takers	O
.	O
</s>
<s>
A	O
deficiency	O
of	O
this	O
analysis	O
is	O
that	O
it	O
does	O
not	O
consider	O
whether	O
the	O
effect	B-Application
size	I-Application
of	O
4	O
points	O
is	O
meaningful	O
.	O
</s>
<s>
If	O
instead	O
of	O
a	O
classroom	O
,	O
we	O
considered	O
a	O
subregion	O
containing	O
900	O
students	O
whose	O
mean	O
score	O
was	O
99	O
,	O
nearly	O
the	O
same	O
z-score	O
and	O
p-value	B-General_Concept
would	O
be	O
observed	O
.	O
</s>
<s>
This	O
shows	O
that	O
if	O
the	O
sample	O
size	O
is	O
large	O
enough	O
,	O
very	O
small	O
differences	O
from	O
the	O
null	O
value	O
can	O
be	O
highly	O
statistically	B-General_Concept
significant	I-General_Concept
.	O
</s>
<s>
Location	B-General_Concept
tests	I-General_Concept
are	O
the	O
most	O
familiar	O
Z-tests	B-General_Concept
.	O
</s>
<s>
Another	O
class	O
of	O
Z-tests	B-General_Concept
arises	O
in	O
maximum	O
likelihood	O
estimation	O
of	O
the	O
parameters	O
in	O
a	O
parametric	B-General_Concept
statistical	O
model	O
.	O
</s>
<s>
The	O
maximum	O
likelihood	O
estimate	O
divided	O
by	O
its	O
standard	B-General_Concept
error	I-General_Concept
can	O
be	O
used	O
as	O
a	O
test	B-General_Concept
statistic	I-General_Concept
for	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
population	O
value	O
of	O
the	O
parameter	O
equals	O
zero	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
is	O
the	O
maximum	O
likelihood	O
estimate	O
of	O
a	O
parameter	O
θ	O
,	O
and	O
θ0	O
is	O
the	O
value	O
of	O
θ	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
</s>
<s>
can	O
be	O
used	O
as	O
a	O
Z-test	B-General_Concept
statistic	O
.	O
</s>
<s>
When	O
using	O
a	O
Z-test	B-General_Concept
for	O
maximum	O
likelihood	O
estimates	O
,	O
it	O
is	O
important	O
to	O
be	O
aware	O
that	O
the	O
normal	O
approximation	O
may	O
be	O
poor	O
if	O
the	O
sample	O
size	O
is	O
not	O
sufficiently	O
large	O
.	O
</s>
<s>
Although	O
there	O
is	O
no	O
simple	O
,	O
universal	O
rule	O
stating	O
how	O
large	O
the	O
sample	O
size	O
must	O
be	O
to	O
use	O
a	O
Z-test	B-General_Concept
,	O
simulation	B-Algorithm
can	O
give	O
a	O
good	O
idea	O
as	O
to	O
whether	O
a	O
Z-test	B-General_Concept
is	O
appropriate	O
in	O
a	O
given	O
situation	O
.	O
</s>
<s>
Z-tests	B-General_Concept
are	O
employed	O
whenever	O
it	O
can	O
be	O
argued	O
that	O
a	O
test	B-General_Concept
statistic	I-General_Concept
follows	O
a	O
normal	O
distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
interest	O
.	O
</s>
<s>
Many	O
non-parametric	B-General_Concept
test	I-General_Concept
statistics	O
,	O
such	O
as	O
U	O
statistics	O
,	O
are	O
approximately	O
normal	O
for	O
large	O
enough	O
sample	O
sizes	O
,	O
and	O
hence	O
are	O
often	O
performed	O
as	O
Z-tests	B-General_Concept
.	O
</s>
