   Six Sigma Glossary from MiC Quality
Sample Size Determination
 Sample Size for a Population Proportion

The sample size to achieve a confidence interval of width 'w' for a large sample can be calculated from: where: the z statistic for the confidence level
w the confidence interval
s the process standard deviation
p the population proportion

The population proportion may not be known before the sample is taken, and so must be estimated.

 Sample Size with Confidence Intervals

The sample size to achieve a confidence interval of width 'w', can be calculated from: where: the z statistic for the confidence level
w the confidence interval
s the process standard deviation

 Sample Size with Hypothesis Tests

Based on a level of significance 'a', and a Type II error 'b' at a departure 'd' from the target mean, the formula for the t-test (small sample sizes) is: where:

ta the t statistic corresponding to the chosen level of significance (use ta/2 for two sided tests)
tb 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: where:

za the z statistic corresponding to the chosen level of significance (use za/2 for two sided tests)
zb the z statistic corresponding to the Type II error (use for both one and two sided tests)

Note that:

• for two sided tests use a/2, but still use b (NOT b/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.
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