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
ttest (small sample sizes) is:
where:
t_{α}
the t statistic corresponding to the chosen
level of significance (use t_{α/2}
for two sided tests)
t_{β}
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:
z_{α} the
z statistic corresponding to the chosen
level of significance (use z_{α/2}
for two sided tests)
z_{Β} 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 tstatistic
depends on the number of degrees of
freedom (n1) the equation is solved
iteratively. Start with the zstatistic
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.
