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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)
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for two sided tests use α/2, but still use β (NOT β/2)
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the standard deviation must be known to use either formula, although a reasonable estimate will serve
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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.
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