[SIX SIGMA GLOSSARY ALPHABETICAL INDEX] [SIX SIGMA GLOSSARY INDEX OF TOPICS]

• Sample Size for a Population Proportion
• Sample Size with Confidence Intervals
• Sample Size with Hypothesis Tests

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.

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

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:

t_a the t statistic corresponding to the chosen level of significance (use t_a/2 for two sided tests)
t_b 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_a the z statistic corresponding to the chosen level of significance (use z_a/2 for two sided tests)

z_b the z statistic corresponding to the Type II error (use for both one and two sided tests)

• 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.