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

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

When statistical tests are planned then it is necessary to decide on the sample size. If this is not sufficiently large then will not be possible to estimate the values of statistical parameters to the required degree of accuracy and level of confidence.

Using excessive sample sizes will not be economical, and the issue of practical vs statistical significance must be considered.

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

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.