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values ⎯x
1
and ⎯x
2
, and standard deviations s
1
and s
2
. If the populations
standard deviations corresponding to the samples, σ
1
and σ
2
, are known, or if
n
1
> 30 and n
2
> 30 (large samples), the test statistic to be used is
If n
1
< 30 or n
2
< 30 (at least one small sample), use the following test statistic:
Two-sided hypothesis
If the alternative hypothesis is a two-sided hypothesis, i.e., H
1
: μ
1
-μ
2
≠δ, The P-
value for this test is calculated as
Θ If using z, P-value = 2⋅UTPN(0,1, |z
o
|)
Θ If using t, P-value = 2⋅UTPT(ν,|t
o
|)
with the degrees of freedom for the t-distribution given by ν = n
1
+ n
2
- 2. The
test criteria are
Θ Reject H
o
if P-value < α
Θ Do not reject H
o
if P-value > α.
One-sided hypothesis
If the alternative hypothesis is a two-sided hypothesis, i.e., H
1
: μ
1
-μ
2
< δ, or,
H
1
: μ
1
-μ
2
< δ,, the P-value for this test is calculated as:
Θ If using z, P-value = UTPN(0,1, |z
o
|)
Θ If using t, P-value = UTPT(ν,|t
o
|)
2
2
2
1
2
1
21
)(
nn
xx
z
o
σσ
δ
+
−−
=
21
2121
2
22
2
11
21
)2(
)1()1(
)(
nn
nnnn
snsn
xx
t
+
−+
−+−
−−
=
δ