unknown
but assumed equal
σ
1
and
σ
2
unknown,
not assumed equal

Chap 10-20
Population means,
independent
samples
σ
1
and
σ
2
known
σ
1
and
σ
2
unknown,
large samples
*
σ
1
and
σ
2
unknown
but assumed equal
σ
1
and
σ
2
unknown,
not assumed equal
Where
t
has (n
1
+ n
2
–
2) d.f.,
and
2
n
n
s
1
n
s
1
n
s
2
1
2
2
2
2
1
1
p
2
1
p
2
1
2
1
n
1
n
1
s
μ
μ
x
x
t
The test statistic for
μ
1
–
μ
2
is:

Chap 10-21
Population means,
independent
samples
σ
1
and
σ
2
known
σ
1
and
σ
2
unknown,
small samples
The test statistic for
μ
1
–
μ
2
is:
*
σ
1
and
σ
2
unknown
but assumed equal
σ
1
and
σ
2
unknown,
not assumed equal
2
2
2
1
2
1
2
1
2
1
n
s
n
s
μ
μ
x
x
t
Where
t
has
d.f. given by
1
n
/n
s
1
n
/n
s
)
/n
s
/n
(s
df
2
2
2
2
2
1
2
1
2
1
2
2
2
2
1
2
1

Chap 10-22
Two Population Means, Independent Samples
Lower tail test:
H
0
:
μ
1
–
μ
2
0
H
A
:
μ
1
–
μ
2
< 0
Upper tail test:
H
0
:
μ
1
–
μ
2
≤
0
H
A
:
μ
1
–
μ
2
> 0
Two-tailed test:
H
0
:
μ
1
–
μ
2
= 0
H
A
:
μ
1
–
μ
2
≠
0
/2
/2
-z
-z
/2
z
z
/2
Reject H
0
if z < -z
Reject H
0
if z > z
Reject H
0
if z < -z
/2
or z > z
/2
Hypothesis tests for
μ
1
–
μ
2
Example:
σ
1
and
σ
2
known: