| Based on a level of significance
'α',
and a Type II error 'β'
at a departure 'δ'
from the target mean, the formula for the
t-test (small sample sizes) is:
where:
tα
the t statistic corresponding to the chosen
level of significance (use tα/2
for two sided tests)
tβ
the t statistic corresponding to the Type
II error (use for both one and two sided
tests)
Note that:
The formula for the z test (large sample
size) is essentially the same. For large
sample sizes the t statistic converges to
the z statistic:
zα the
z statistic corresponding to the chosen
level of significance (use zα/2
for two sided tests)
zΒ the
z statistic corresponding to the Type
II error (use for both one and two sided
tests)
-
for two sided tests
use α/2,
but still use β
(NOT
β/2)
-
the standard deviation
must be known to use either formula,
although a reasonable estimate will
serve
-
because the t-statistic
depends on the number of degrees of
freedom (n-1) the equation is solved
iteratively. Start with the z-statistic
and find an approximation for 'n' (or
guess n). Use this value of 'n' to find
the t statistic and recalculate to get
a better approximation for 'n'. Repeat
until the values converge.
|
